Abstract
The appendix contains a summary of topological and functional analysis results in reference to the description of the topology and equivalent characterizations of convergence in spaces of test functions and in spaces of distributions. In addition, a variety of foundational results from calculus, measure theory, and special functions originating outside the scope of this book are included here.
Keywords
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13.1 Summary of Topological and Functional Analytic
A topology on a set X is a family τ of subsets of X satisfying the following properties:
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1.
\(X,\varnothing \in \tau\);
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2.
If {A α } α ∈ I is a family of sets contained in τ, then \(\bigcup _{\alpha \in I}A_{\alpha } \in \tau\);
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3.
If \(A_{1},A_{2} \in \tau\), then \(A_{1} \cap A_{2} \in \tau\).
If X and τ are as above, the pair (X, τ) is called a topological space and the elements of τ are called open sets. If x is an element of X (sometimes referred to as point), then any open set containing x is called an open neighborhood of x. Any set that contains an open neighborhood of x is called a neighborhood of x. A sequence \(\{x_{j}\}_{j\in \mathbb{N}}\) of elements of a topological space (X, τ) are said to converge to x ∈ X, and we write \(\lim \limits _{j\rightarrow \infty }x_{j} = x\), if every neighborhood of x contains all but a finite number of the elements of the sequence.
A family \(\mathcal{B}\) of subsets of X is a base for a topology τ on X if \(\mathcal{B}\) is a subfamily of τ and for each x ∈ X, and each neighborhood U of x, there is \(V \in \mathcal{B}\) such that \(x \in V \subset U\). We say that the base \(\mathcal{B}\) generates the topology τ. An equivalent characterization of bases that is useful in applications is as follows. A base for a topological space (X, τ) is a collection \(\mathcal{B}\) of open sets in τ such that every open set in τ can be written as a union of elements of \(\mathcal{B}\).
It is important to note that not any family of subsets of a given set is a base for some topology on the set. Given a set X and a family \(\mathcal{B}\) of subsets of X, this family \(\mathcal{B}\) is a base for some topology on X if and only if
If τ 1 and τ 2 are two topologies on a set X, such that every member of τ 2 is also a member of τ 1, then we say that τ 1 is finer (or, larger) than τ 2, and that τ 2 is coarser (or, smaller) than τ 1. If (X, τ) is a topological space and \(E \subseteq X\), then the topology induced by τ on E is the topology for which all open sets are intersections of open sets in τ with E.
Let (X, τ) and (Y, τ ′) be two topological spaces. The topology on X ×Y with base the collection of products of open sets in X and open sets in Y [which satisfies (13.1.1)] is called the product topology. A function f: X → Y is called continuous at x ∈ X if the inverse image under f of every open neighborhood of f(x) is an open neighborhood of x. It is called continuous on X if it is continuous at every x ∈ X. In particular, it is easy to see that if f: X → Y is continuous then \(\lim \limits _{j\rightarrow \infty }f(x_{j}) = f(x)\) for every convergent sequence \(\{x_{j}\}_{j\in \mathbb{N}}\) of X converging to x ∈ X, that is, f is sequentially continuous. While in general the converse is false, in the case when X is in fact a metric space (see below), if f: X → Y is sequential continuous, then f is continuous.
A topological space (X, τ) is said to be separated, or Hausdorff, if for any two distinct elements x and y of X, there exist U, a neighborhood of x and V, a neighborhood of y, such that \(U \cap V = \varnothing \). In a Hausdorff space the limit of a convergent sequence is unique.
A metric space is a set X equipped with a distance function (also called metric). This is a function d: X ×X → [0, ∞) satisfying the following properties:
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1.
If x, y ∈ X, then d(x, y) = 0 if and only if x = y (nondegeneracy);
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2.
d(x, y) = d(y, x) for every x, y ∈ X (symmetry);
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3.
d(x, y) ≤ d(x, z) + d(z, y) for every x, y, z ∈ X (triangle inequality).
Let d be a metric on X. The open balls B(x, r): = { y ∈ X: d(x, y) < r}, x ∈ X, r ∈ (0, ∞), are then the base of a Hausdorff topology on X (since it satisfies (13.1.1)), denoted τ d . For a sequence \(\{x_{j}\}_{j\in \mathbb{N}}\) of points in X and x ∈ X one has \(\lim \limits _{j\rightarrow \infty }x_{j} = x\) in the topology τ d , if and only if the sequence of numbers \(\{d(x_{j},x)\}_{j\in \mathbb{N}}\) converges to 0 as j → ∞. A sequence \(\{x_{j}\}_{j\in \mathbb{N}}\) of points in (X, τ d ) is called Cauchy if \(d(x_{j},x_{k}) \rightarrow 0\) as j, k → ∞. A metric space is called complete if every Cauchy sequence is convergent.
A topological space X is called metrizable if there exists a distance function for which the open balls form a base for the topology (i.e., the topology on X coincides with τ d ).
A vector space X over \(\mathbb{C}\) becomes a topological vector space if equipped with a topology that is compatible with its vector structure, that is, the operation of vector addition, and also the operation of multiplication by a complex number, are continuous maps X ×X → X and \(\mathbb{C} \times X \rightarrow X\), respectively (where X ×X and \(\mathbb{C} \times X\) are endowed each with the corresponding product topology). Observe that in order to specify a topology on a vector space, it suffices to give the system of neighborhoods of the zero element 0 ∈ X, since the system of neighborhoods of any other element of X is obtained from this via translation. In fact, it suffices to give a base of neighborhoods of 0. This is a family \(\mathcal{B}\) of neighborhoods of 0 such that every neighborhood of 0 contains a member of \(\mathcal{B}\). A set \(E \subseteq X\) is then open if and only if, for every x ∈ E, there exists \(U \in \mathcal{B}\) such that \(x + U \subseteq E\), where \(x + U:=\{ x + y:\, y \in U\}\).
A topological vector space is called locally convex if it has a base of neighborhoods of 0 consisting of sets that are convex, balanced, and absorbing. A set U is called convex provided \(tx + (1 - t)y \in U\) whenever x, y ∈ U and t ∈ [0, 1]. A set U is called balanced if cx ∈ U whenever x ∈ U, \(c \in \mathbb{C}\) and | c | ≤ 1. A set U is called absorbing provided for each x ∈ X there exists t > 0 such that x ∈ tU: = { ty: y ∈ U}.
A seminorm on a vector space X is a function \(p: X \rightarrow \mathbb{R}\) satisfying the properties
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1.
p(cx) = | c | p(x), for every \(c \in \mathbb{C}\) and every x ∈ X (positively homogeneous);
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2.
\(p(x + y) \leq p(x) + p(y)\), for every x, y ∈ X (sub-additive).
In particular, a seminorm p on X satisfies p(0) = 0 and p(x) ≥ 0 for every x ∈ X.
A family \(\mathcal{P}\) of seminorms on X is called separating if, for each x ∈ X, x ≠ 0, there exists \(p \in \mathcal{P}\) such that p(x) ≠ 0. Given a separating family of seminorms \(\mathcal{P}\) on X, let B be the collection of sets of the form
Then B is a base of neighborhoods of 0 of a locally convex vector space topology \(\tau _{\mathcal{P}}\) on X called the topology generated by the family of seminorms \(\mathcal{P}\). In this context, it may be readily verified that if Y is a linear subspace of X and \(\mathcal{P}\vert _{Y }:=\{ p\vert _{Y }:\, p \in \mathcal{P}\}\), then
Conversely, if X is a locally convex topological vector space, for each U convex, balanced, and absorbing neighborhood of 0, the mapping \(p_{U}: X \rightarrow \mathbb{R}\) defined by \(p_{U}(x):=\inf \{ t > 0:\, {t}^{-1}x \in U\}\), x ∈ X (called the Minkowski functional associated with U) is a seminorm. It is then not hard to see that the topology on X is generated by this family of seminorms (in the manner described above).
Let \(\mathcal{P} =\{ p_{j}\}_{j\in \mathbb{N}}\) be a countable family of seminorms that is also separating (thus, if x ∈ X and p j (x) = 0 for all \(j \in \mathbb{N}\), then x = 0). The topology generated by this family is metrizable. Indeed, the function \(d: X \times X \rightarrow \mathbb{R}\) defined by
is a distance on X and the topology τ d induced by the metric d coincides with the topology generated by \(\mathcal{P}\). In the converse direction, it can be shown that the topology of a locally convex space that is metrizable, endowed with a translation invariant metric, can be generated by a countable family of seminorms.
A locally convex topological vector space that is metrizable and complete is called a Fréchet space. Thus, if a family \(\mathcal{P} =\{ p_{j}\}_{j\in \mathbb{N}}\) of seminorms generates the topology of a Fréchet space X, then whenever \(p_{j}(x_{k} - x_{l}) \rightarrow 0\) as k, l → ∞ for every \(j \in \mathbb{N}\), there exists x ∈ X such that \(p_{j}(x_{k} - x) \rightarrow 0\) as k → ∞ for every \(j \in \mathbb{N}\).
Let X be a vector space, {X j } j ∈ J be a family of vector subspaces of X such that \(X =\bigcup \limits _{j\in J}X_{j}\) and if \(j_{1},j_{2} \in J\) then there exists j 3 ∈ J with \(X_{j_{1}} \subset X_{j_{3}}\) and \(X_{j_{2}} \subset X_{j_{3}}\). Also assume that there exist topologies τ j on X j such that if \(X_{j_{1}} \subset X_{j_{2}}\); then the topology \(\tau _{j_{1}}\) is finer than the topology \(\tau _{j_{2}j_{1}}\) induced on \(X_{j_{1}}\) by \(X_{j_{2}}\). Let
Then \(\mathcal{W}\) is a base of neighborhoods of 0 in a locally convex topology τ on X. Call this topology the inductive limit topology on X.
If in addition whenever \(X_{j_{1}} \subset X_{j_{2}}\) we also have \(\tau _{j_{2}j_{1}} =\tau _{j_{1}}\), we call the inductive limit strict. If the topology τ on X is the strict inductive limit of the topologies of an increasing sequence \(\{X_{n}\}_{n\in \mathbb{N}}\), then the topology induced on X n by the topology τ on X coincides with the initial topology τ n on X n , for every \(n \in \mathbb{N}\).
In general, if X is a topological vector space, its dual space is the collection of all linear mappings \(f: X \rightarrow \mathbb{C}\) (also referred to as functionals) that are continuous with respect to the topology on X. In the case when X is a locally convex topological vector space and \(\mathcal{P}\) is a family of seminorms generating the topology of X, then a seminorm q on X is continuous if and only if there exist \(N \in \mathbb{N}\), seminorms \(p_{1},\ldots,p_{N} \in \mathcal{P}\), and a constant C ∈ (0, ∞), such that
This fact then gives a criterion for continuity for functionals on X, since if \(f: X \rightarrow \mathbb{C}\) is a linear mapping, then q(x): = | f(x) | , for x ∈ X, is a seminorm on X. In addition, if the family of seminorms \(\mathcal{P}\) has the property that for any p 1, \(p_{2} \in \mathcal{P}\) there exists \(p_{3} \in \mathcal{P}\) with the property that \(\max \{p_{1}(x),p_{2}(x)\} \leq p_{3}(x)\) for every x ∈ X, then a linear functional \(f: X \rightarrow \mathbb{C}\) is continuous if and only if there exist a seminorm \(p \in \mathcal{P}\) and a constant C ∈ (0, ∞), such that
Given a nonempty set X and a family \(\mathcal{F}\) of mappings \(f: X \rightarrow \mathbb{C}\), denote by \(\tau _{\mathcal{F}}\) the collection of all unions of finite intersections of sets f − 1(V ), for \(f \in \mathcal{F}\) and V open set in \(\mathbb{C}\). Then \(\tau _{\mathcal{F}}\) is a topology on X, and is the weakest topology on X that makes every \(f \in \mathcal{F}\) continuous. We will refer to it as the \(\mathcal{F}\) -topology on X.
Let X be a vector space, \(\mathcal{F}\) be a separating vector space of linear functionals on X and \(\tau _{\mathcal{F}}\) be the \(\mathcal{F}\)-topology on X. Then \((X,\tau _{\mathcal{F}})\) is a locally convex topological space and its dual is \(\mathcal{F}\). In particular, a sequence {x j } j in X satisfies x j → 0 in \(\tau _{\mathcal{F}}\) as j → ∞ if and only if f(x j ) → 0 as j → ∞ for every \(f \in \mathcal{F}\).
If X is a topological vector space and X ′ denotes its dual space, then every x ∈ X induces a linear functional F x on X ′ defined by F x (Λ): = Λ(x) for every Λ ∈ X ′ and if we set \(\mathcal{F}:=\{ F_{x}:\, x \in X\}\), then \(\mathcal{F}\) separates points in X ′. In particular, the \(\mathcal{F}\)-topology on X ′, called the weak ∗ -topology on X ′, is locally convex and every linear functional on X ′ that is continuous with respect to the weak ∗ -topology is precisely of the form F x , for some x ∈ X. Moreover, a sequence \(\{\Lambda _{j}\}_{j} \subset X^{\prime}\) converges to some Λ ∈ X ′, weak ∗ -topology on X ′, if and only if Λ j (x) → Λ(x) as j → ∞ for every x ∈ X.
An inspection of the definition of the weak ∗ -topology yields a description of the open sets in this topology. More precisely, if X is a topological vector space and for \(A \subseteq X\) and \(\varepsilon \in (0,\infty )\) we set
then the following equivalence holds:
The transpose of any linear and continuous operator between two topological vector spaces is always continuous at the level of dual spaces equipped with weak ∗ -topologies.
Proposition 13.1.
Assume X and Y are two given topological vector spaces, and denote by X′, Y ′ their duals, each endowed with the corresponding weak∗-topology. Also, suppose T: X → Y is a linear and continuous operator, and define its transpose T t as the mapping
Then T t is well-defined, linear, and continuous.
Proof.
For each y ′ ∈ Y ′ it follows that y ′ ∘ T is a composition of two linear and continuous mappings. Hence, y ′ ∘ T ∈ X ′ which proves that T t: Y ′ → X ′ is well-defined. It is also clear from (13.1.10) that T t is linear. There remains to prove that T t is continuous. By linearity it suffices to check that T t is continuous at 0. With this goal in mind, fix an arbitrary finite subset A of X along with an arbitrary number \(\varepsilon \in (0,\infty )\), and define
Furthermore, introduce \(\widetilde{A}:=\{ Tx:\, x \in A\}\) and set
Then \({T}^{t}(\widetilde{\mathcal{O}}_{\widetilde{A},\varepsilon }) \subseteq \mathcal{O}_{A,\varepsilon }\). Invoking the description from (13.1.9) we may conclude that T t is continuous at 0, and this finishes the proof of the proposition.
Proposition 13.2.
Suppose that X, Y, Z are topological vector spaces, and denote by X′, Y ′, Z′ their duals, each endowed with the corresponding weak∗-topology. In addition, assume that T: X → Y and R: Y → Z are two linear and continuous operators. Then
In particular, if T: X → Y is a linear, continuous, bijective map, with continuous inverse T −1 : Y → X, then T t : Y ′→ X′ is also bijective and has a continuous inverse (T t ) −1 : X′→ Y ′ that satisfies \({({T}^{t})}^{-1} = {({T}^{-1})}^{t}\) .
Proof.
Formula (13.1.13) is immediate from definitions, while the claims in the last part of the statement are direct consequences of (13.1.13) and the fact that the transpose of the identity is also the identity.
We also state and prove an embedding result at the level of dual spaces endowed with the weak ∗ -topology.
Proposition 13.3.
Suppose X and Y are topological vector spaces such that X ⊆ Y densely and the inclusion map \(\iota: X \rightarrow Y\), \(\iota (x):= x\) for each x ∈ X, is continuous. Then Y ′ endowed with the weak∗-topology embeds continuously in the space X′ endowed with the weak∗-topology, in the sense that the mapping
is well-defined, linear, injective and continuous. Under the identification of Y ′ with \({\iota }^{t}(Y \prime) \subseteq X^{\prime}\) , we therefore have
Proof.
The fact that \({\iota }^{t}\) is well-defined, linear, and continuous follows directly from Proposition 13.1 and assumptions. Assume now that y ′ ∈ Y ′ is such that \({\iota }^{t}(y^{\prime}) = 0\). Then from the fact that \(y^{\prime}\circ \iota: X \rightarrow \mathbb{C}\) is zero we deduce that \(y^{\prime}\big\vert _{X} = 0\). Since \(y^{\prime}: Y \rightarrow \mathbb{C}\) is continuous and X is dense in Y, we necessarily have that y ′ vanishes on Y, forcing y ′ = 0 in Y ′. Keeping in mind that \({\iota }^{t}\) is linear, this implies that \({\iota }^{t}\) is injective.
Theorem 13.4 (Hahn–Banach Theorem).
Let X be a vector space (over complex numbers) and suppose p: X → [0,∞) is a seminorm on X. Also assume that Y is a linear subspace of X and that \(\varphi: Y \rightarrow \mathbb{C}\) is a linear functional dominated by p on Y, that is,
Then there exists a linear functional \(\Phi : X \rightarrow \mathbb{C}\) satisfying
In the next installment we shall specialize the above considerations to various specific settings used in the book. The reader is reminded that \(\Omega \subseteq {\mathbb{R}}^{n}\) denotes a fixed, nonempty, arbitrary open set.
The Topological Vector Space \(\mathcal{E}(\Omega )\) By τ we denote the topology on C ∞(Ω) generated by the following family of seminorms:
where \(K \subset \Omega \) is a compact set and \(m \in \mathbb{N}_{0}\). Consequently, a sequence \(\varphi _{j} \in {C}^{\infty }(\Omega )\), \(j \in \mathbb{N}\), converges in τ to a function \(\varphi \in {C}^{\infty }(\Omega )\) as j → ∞, if and only if for any compact set \(K \subset \Omega \) and any \(m \in \mathbb{N}_{0}\) one has
We will use the notation
The space \(\mathcal{E}(\Omega )\) is locally convex and metrizable since its topology is defined by the family of countable seminorms \(\{p_{K_{m},m}\}_{m\in \mathbb{N}_{0}}\) where
In addition, τ is independent of the family \(\{K_{m}\}_{m\in \mathbb{N}_{0}}\) with the above properties and \(\mathcal{E}(\Omega )\) is complete. Thus, \(\mathcal{E}(\Omega )\) is a Frechét space.
The Topological Vector Space \(\mathcal{E}^{\prime}(\Omega )\) Based on the discussion on dual spaces for locally convex topological vector spaces (see (13.1.7) and the remarks preceding it), it follows that the dual space of \(\mathcal{E}(\Omega )\) is the collection of all linear functionals \(u: \mathcal{E}(\Omega ) \rightarrow \mathbb{C}\) for which there exist \(m \in \mathbb{N}\), a compact set \(K \subset {\mathbb{R}}^{n}\) with K ⊂ Ω, and a constant C > 0 such that
This dual space will be endowed with the weak ∗ -topology induced by \(\mathcal{E}(\Omega )\) and we denote this topological space by \(\mathcal{E}^{\prime}(\Omega )\). Hence, if for each \(\varphi \in {C}^{\infty }(\Omega )\) we consider the evaluation mapping \(F_{\varphi }\) taking any functional u from the dual of \(\mathcal{E}(\Omega )\) into the number \(F_{\varphi }(u):= u(\varphi ) \in \mathbb{C}\), then the family \(\mathcal{F}:=\{ F_{\varphi }:\,\varphi \in \mathcal{E}(\Omega )\}\) separates points in the dual of \(\mathcal{E}(\Omega )\), and the weak ∗ -topology on this dual is the \(\mathcal{F}\)-topology on it. In particular, if \(\{u_{j}\}_{j\in \mathbb{N}}\) is a sequence in \(\mathcal{E}^{\prime}(\Omega )\) and \(u \in \mathcal{E}^{\prime}(\Omega )\), then
Moreover, a sequence \(\{u_{j}\}_{j\in \mathbb{N}}\) in \(\mathcal{E}^{\prime}(\Omega )\) is Cauchy provided \(\lim \limits _{j,k\rightarrow \infty }(u_{j}-u_{k})(\varphi )\! =\! 0\) for every \(\varphi \in {C}^{\infty }(\Omega )\) and the weak ∗ -topology on the dual of \(\mathcal{E}(\Omega )\) is locally convex and complete.
The Topological Vector Space \(\mathcal{D}_{K}(\Omega )\) Let \(K \subseteq \Omega \) be a compact set in \({\mathbb{R}}^{n}\). Denote by \(\mathcal{D}_{K}(\Omega )\) the topological vector space of functions \(\{f \in {C}^{\infty }(\Omega ):\, \mbox{ supp}\,f \subseteq K\}\) with the topology induced by τ, the topology in \(\mathcal{E}(\Omega )\). Then the topology on \(\mathcal{D}_{K}(\Omega )\) is generated by the family of seminorms
where \(m \in \mathbb{N}_{0}\). Hence, \(\mathcal{D}_{K}(\Omega )\) is a Fréchet space. In addition, a linear mapping \(u: \mathcal{D}_{K}(\Omega ) \rightarrow \mathbb{C}\) is continuous if and only if there exist \(m \in \mathbb{N}\) and a constant C > 0, both depending on K, such that
The Topological Vector Space \(\mathcal{D}(\Omega )\) The topological vector space on \(C_{0}^{\infty }(\Omega )\) endowed with the inductive limit topology of the Frechét spaces \(\mathcal{D}_{K}(\Omega )\) will be denoted by \(\mathcal{D}(\Omega )\). In this setting, we have
The topology \(\mathcal{D}(\Omega )\) is locally convex and complete but not metrizable (thus not Fréchet) and is the strict inductive limit of the topologies \(\{\mathcal{D}_{K_{j}}(\Omega )\}_{j\in \mathbb{N}}\), where the family \(\{K_{j}\}_{j\in \mathbb{N}}\) is as in (13.1.21).
We also record an important result that is proved in [59, Theorem 6.6, p. 155].
Theorem 13.5.
Let X be a locally convex topological vector space and suppose the map \(\Lambda : \mathcal{D}(\Omega ) \rightarrow X\) is linear. Then Λ is continuous if and only if for every sequence \(\{\varphi _{j}\}_{j\in \mathbb{N}}\) in \(C_{0}^{\infty }(\Omega )\) satisfying \(\varphi _{j}\mathop{\longrightarrow}\limits_{j \rightarrow \infty }^{\mathcal{D}(\Omega )}0\) we have \(\lim \limits _{j\rightarrow \infty }\Lambda (\varphi _{j}) = 0\) in X.
The Topological Vector Space \(\mathcal{D}^{\prime}(\Omega )\) The dual space of \(\mathcal{D}(\Omega )\) endowed with the weak ∗ -topology is denoted by \(\mathcal{D}^{\prime}(\Omega )\). Hence, if \(\{u_{j}\}_{j\in \mathbb{N}}\) is a sequence in \(\mathcal{D}^{\prime}(\Omega )\) and \(u \in \mathcal{D}^{\prime}(\Omega )\), then
In addition, a sequence \(\{u_{j}\}_{j\in \mathbb{N}}\) in \(\mathcal{D}^{\prime}(\Omega )\) is called Cauchy provided
for every \(\varphi \in C_{0}^{\infty }(\Omega )\). The weak ∗ -topology on the dual of \(\mathcal{D}(\Omega )\) is locally convex and complete and an inspection of this topology reveals that it coincides with the topology defined by the family of seminorms
The Topological Vector Space \(\mathcal{S}({\mathbb{R}}^{n})\) The Schwartz class of rapidly decreasing functions is the vector space
endowed with the topology generated by the family of seminorms \(\{p_{k,m}\}_{k,m\in \mathbb{N}_{0}}\) defined by
Hence, the topology generated by the family of seminorms \(\{p_{k,m}\}_{k,m\in \mathbb{N}_{0}}\) on \(\mathcal{S}({\mathbb{R}}^{n})\) is locally convex, metrizable, and since it is also complete, the space \(\mathcal{S}({\mathbb{R}}^{n})\) is Frechét. Moreover, a sequence \(\varphi _{j} \in \mathcal{S}({\mathbb{R}}^{n})\), \(j \in \mathbb{N}\), converges in \(\mathcal{S}({\mathbb{R}}^{n})\) to a function \(\varphi \in \mathcal{S}({\mathbb{R}}^{n})\) as j → ∞, if and only if for every \(m,k \in \mathbb{N}_{0}\) one has
The Topological Vector Space \(\mathcal{S}^{\prime}({\mathbb{R}}^{n})\) By the discussion about dual spaces for locally convex topological vector spaces, it follows that the dual space of \(\mathcal{S}({\mathbb{R}}^{n})\) is the collection of all linear functions \(u: \mathcal{S}({\mathbb{R}}^{n}) \rightarrow \mathbb{C}\) for which there exist \(m,k \in \mathbb{N}_{0}\), and a finite constant C > 0, such that
We endow the dual of \(\mathcal{S}({\mathbb{R}}^{n})\) with the weak ∗ -topology and denote the resulting locally convex topological vector space by \(\mathcal{S}^{\prime}({\mathbb{R}}^{n})\). Hence, if \(\{u_{j}\}_{j\in \mathbb{N}}\) is a sequence in \(\mathcal{S}^{\prime}({\mathbb{R}}^{n})\) and \(u \in \mathcal{S}^{\prime}({\mathbb{R}}^{n})\), then
Also, a sequence \(\{u_{j}\}_{j\in \mathbb{N}}\) in \(\mathcal{S}^{\prime}({\mathbb{R}}^{n})\) is called Cauchy provided
13.2 Summary of Basic Results from Calculus, Measure Theory, and Topology
Proposition 13.6 (Multinomial theorem).
If \(x = (x_{1},\ldots,x_{n}) \in {\mathbb{R}}^{n}\) and \(N \in \mathbb{N}\) are arbitrary, then
Theorem 13.7 (Binomial theorem).
For any \(x,y \in {\mathbb{C}}^{n}\) and any \(\gamma \in \mathbb{N}_{0}^{n}\) we have
(with the convention that z 0 := 1 for each \(z \in \mathbb{C}\) ).
In the particular case when \(x = (1,\ldots,1) \in {\mathbb{C}}^{n}\) and \(y = (-1,\ldots,-1) \in {\mathbb{C}}^{n}\), formula (13.2.2) yields
Proposition 13.8 (Leibniz’s formula).
Suppose that \(U \subseteq {\mathbb{R}}^{n}\) is an open set, \(N \in \mathbb{N}\) , and \(f,g: U \rightarrow \mathbb{C}\) are two functions of class C N in U. Then
for every multi-index \(\alpha \in \mathbb{N}_{0}^{n}\) of length ≤ N.
It is useful to note that for each \(\alpha,\beta \in \mathbb{N}_{0}^{n}\) and \(x \in {\mathbb{R}}^{n}\),
Theorem 13.9 (Taylor’s formula).
Assume \(U \subseteq {\mathbb{R}}^{n}\) is an open convex set, and that \(N \in \mathbb{N}\) . Also, suppose that \(f: U \rightarrow \mathbb{C}\) is a function of class C N+1 on U. Then for every x,y ∈ U one has
In particular, for each x,y ∈ U there exists θ ∈ (0,1) with the property that
Theorem 13.10 (Rademacher’s theorem).
If \(f: {\mathbb{R}}^{n} \rightarrow \mathbb{R}\) is a Lipschitz function with Lipschitz constant less than or equal to M, for some M ∈ (0,∞), then f is differentiable almost everywhere and \(\|\partial _{k}f\|_{{L}^{\infty }({\mathbb{R}}^{n})} \leq M\) for each \(k = 1,\ldots,n\) .
Theorem 13.11 (Lebesgue’s differentiation theorem).
If \(f \in L_{\mathrm{loc}}^{1}({\mathbb{R}}^{n})\) , then
In particular,
Theorem 13.12 (Lebesgue’s dominated convergence theorem).
Let (X,μ) be a positive measure space and assume that g ∈ L 1 (X,μ) is a nonnegative function. If \(\{f_{j}\}_{j\in \mathbb{N}}\) is a sequence of μ-measurable, complex valued functions on X, such that |f j (x)|≤ g(x) for μ-almost every x ∈ X and \(f(x):=\lim \limits _{j\rightarrow \infty }f_{j}(x)\) exists (in \(\mathbb{C}\) ) for μ-almost every x ∈ X, then f ∈ L 1 (X,μ) and \(\lim \limits _{j\rightarrow \infty }\int _{X}\vert f_{j} - f\vert \,\mathrm{d}\mu = 0\) . In particular, \(\lim \limits _{j\rightarrow \infty }\int _{X}f_{j}\,\mathrm{d}\mu =\int _{X}f\,\mathrm{d}\mu\) .
Theorem 13.13 (Young’s inequality).
Assume that \(1 \leq p,q,r \leq \infty \) are such that \(\frac{1} {p} + \frac{1} {q} = \frac{1} {r} + 1\) . Then for every \(f \in {L}^{p}({\mathbb{R}}^{n})\), \(g \in {L}^{q}({\mathbb{R}}^{n})\) it follows that f ∗ g is well-defined almost everywhere in \({\mathbb{R}}^{n}\), \(f {\ast} g \in {L}^{r}({\mathbb{R}}^{n})\) , and \(\|f {\ast} g\|_{{L}^{r}({\mathbb{R}}^{n})} \leq \| f\|_{{L}^{p}({\mathbb{R}}^{n})}\|g\|_{{L}^{q}({\mathbb{R}}^{n})}\) .
Theorem 13.14.
Let X and Y be two Hausdorff topological spaces.
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(a)
If Λ: X → Y is continuous, then Λ is sequentially continuous.
-
(b)
If Λ: X → Y is sequentially continuous and X is metrizable, then Λ is continuous.
For a proof of Theorem 13.14 see [59, Theorem, p. 395].
Definition 13.15.
A rigid transformation, or isometry, of the Euclidean space \({\mathbb{R}}^{n}\) is any distance preserving mapping of \({\mathbb{R}}^{n}\), that is, any function \(T: {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}\) satisfying
A rigid transformation of \({\mathbb{R}}^{n}\) is any distance preserving mapping of \({\mathbb{R}}^{n}\), i.e., any function \(T: {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}\) satisfying \(\vert Tx - Ty\vert = \vert x - y\vert \) for every \(x,y \in {\mathbb{R}}^{n}\). The rigid transformations of the Euclidean space \({\mathbb{R}}^{n}\) are precisely those obtained by composing a translation with a mapping in \({\mathbb{R}}^{n}\) given by an orthogonal matrix. In other words, a mapping \(T: {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}\) is a rigid transformation of \({\mathbb{R}}^{n}\) if and only if there exist \(x_{0} \in {\mathbb{R}}^{n}\) and an orthogonal matrix \(A: {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}\) with the property that
For a proof of the following version of the Arzelà–Ascoli theorem see [57, Corollary 34, p. 179].
Theorem 13.16 (Arzelà–Ascoli theorem).
Let \(\mathcal{F}\) be an equicontinuous family of real-valued functions on a separable space X. Then each sequence \(\{f_{j}\}_{j\in \mathbb{N}}\) in \(\mathcal{F}\) which is bounded at each point has a subsequence \(\{f_{j_{k}}\}_{k\in \mathbb{N}}\) that converges pointwise to a continuous function, the converges being uniform on each compact subset of X.
Theorem 13.17 (Riesz’s representation theorem for positive functionals).
Let X be a locally compact Hausdorff topological space and Λ a positive linear functional on the space of continuous, compactly supported functions on X (denoted by C 0 0 (X)). Then there exists a unique σ-algebra \(\mathfrak{M}\) on X, which contains all Borel sets on X, and a unique measure \(\mu: \mathfrak{M} \rightarrow [0,\infty ]\) that represents Λ, that is, the following hold:
-
(i)
Λf = ∫ X f d μ for every continuous, compactly supported function f on X;
-
(ii)
μ(K) < ∞ for every compact K ⊂ X;
-
(iii)
For every \(E \in \mathfrak{M}\) we have \(\mu (E) =\inf \{\mu (V ):\, E \subset V,\,V \mbox{ open}\}\) ;
-
(iv)
\(\mu (E) =\sup \,\{\mu (K):\, K \subset E,\,K\mbox{ compact}\}\) for every open set E and every \(E \in \mathfrak{M}\) with μ(E) < ∞;
-
(v)
If \(E \in \mathfrak{M}\) , A ⊂ E, and μ(E) = 0, then μ(A) = 0.
Theorem 13.18 (Riesz’s representation theorem for complex functionals).
Let X be a locally compact Hausdorff topological space and consider the space of continuous functions on X vanishing at infinity, that is,
Then C oo (X) is the closure in the uniform norm of \(C_{0}^{0}(X)\) and for every bounded linear functional \(\Lambda : C_{oo}(X) \rightarrow \mathbb{C}\) there exists a unique regular complex Borel measure μ on X such that \(\Lambda f =\int _{X}f\,\mathrm{d}\mu\) for every f ∈ C oo (X) and \(\|\Lambda \| = \vert \mu \vert (X)\) .
Theorem 13.19 (Riesz’s representation theorem for locally bounded functionals).
Let X be a locally compact Hausdorff topological space and assume that \(\Lambda : C_{0}^{0}(X) \rightarrow \mathbb{R}\) is a linear functional that is locally bounded, in the sense that for each compact set \(K \subset X\) there exists a constant C K ∈ (0,∞) such that
Then there exist two measures μ 1 ,μ 2 , taking Borel sets from X into [0,∞], and satisfying properties (ii)–(iv) in Theorem 13.17, such that
The reader is warned that since both μ 1 and μ 2 are allowed to take the value ∞, their difference \(\mu _{1} -\mu _{2}\) is not well-defined in general. This being said, \(\mu _{1} -\mu _{2}\) is a well-defined finite signed measure on each compact subset of X.
Proposition 13.20 (Urysohn’s lemma).
If X is a locally compact Hausdorff space and \(K \subset U \subset X\) are such that K is compact and U is open, then there exists a function \(f \in C_{0}^{0}(U)\) that satisfies f = 1 on K and 0 ≤ f ≤ 1.
Theorem 13.21 (Vitali’s convergence theorem).
Let (X,μ) be a positive measure space with μ(X) < ∞. Suppose \(\{f_{k}\}_{k\in \mathbb{N}}\) is a sequence of functions in L 1 (X,μ) and that f is a function on X (all complex-valued) satisfying:
-
(i)
f k (x) → f(x) for μ-almost every x ∈ X as k →∞;
-
(ii)
|f| < ∞ μ-almost everywhere in X;
-
(iii)
\(\{f_{k}\}_{k\in \mathbb{N}}\) is uniformly integrable, in the sense that for every \(\varepsilon > 0\) there exists δ > 0 such that for every \(k \in \mathbb{N}\) we have \(\Big\vert \int _{E}f_{k}\,\mathrm{d}\mu \Big\vert <\varepsilon\) whenever \(E \subseteq X\) is a μ-measurable set with μ(E) < δ.
Then f ∈ L 1 (X,μ) and
In particular, \(\lim \limits _{k\rightarrow \infty }\int _{X}f_{k}\,\mathrm{d}\mu =\int _{X}f\,\mathrm{d}\mu\) .
See, for example, [58, p. 133].
Proposition 13.22.
Let (X,μ) be a positive measure space and suppose that f ∈ L 1 (X,μ). Then for every \(\varepsilon > 0\) there exists δ > 0 such that for every μ-measurable set \(A \subseteq X\) satisfying μ(A) < δ we have \(\int _{A}\vert f\vert \,\mathrm{d}\mu <\varepsilon\) .
Proof.
Consider the measure λ: = | f | μ on X. Then λ is absolutely continuous with respect to μ and the \((\varepsilon,\delta )\) characterization of absolute continuity of measures (see, e.g., [58, Theorem 6.11, p. 124]) yields the desired conclusion.
13.3 Custom-Designing Smooth Cut-Off Functions
Lemma 13.23.
Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be the function defined by
Then f is of class C ∞ on \(\mathbb{R}\) .
Proof.
Denote by \(\mathcal{C}\) the collection of functions \(g: \mathbb{R} \rightarrow \mathbb{R}\) for which there exists a polynomial P such that
Recall that if \(h: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function that is differentiable on \(\mathbb{R} \setminus \{ 0\}\) and for which there exists \(L \in \mathbb{R}\) such that \(\lim \limits _{x\rightarrow {0}^{-}}h\prime(x) = L =\lim \limits _{x\rightarrow {0}^{+}}h\prime(x)\), then h is also differentiable at the origin and h ′(0) = L. An immediate consequence of this fact is that any \(g \in \mathcal{C}\) is differentiable and \(g\prime \in \mathcal{C}\). In turn, this readily gives that any \(g \in \mathcal{C}\) is of class C ∞ on \(\mathbb{R}\). Since f defined in (13.3.1) clearly belongs to \(\mathcal{C}\), it follows that f is of class C ∞ on \(\mathbb{R}\).
Lemma 13.24.
The function \(\phi: {\mathbb{R}}^{n} \rightarrow \mathbb{R}\) defined by
where \(C:={\left (\omega _{n-1}\int _{0}^{\infty }{\mathrm{e}{}^{1/{(\rho }^{2}-1) }\rho }^{n-1}\,\mathrm{d}\rho \right)}^{-1} \in (0,\infty )\) , satisfies the following properties:
Proof.
That ϕ ≥ 0 and \(\mathrm{supp}\,\phi \subseteq \overline{B(0,1)}\) is immediate from its definition. Also, since \(\phi (x) = f(1 -\vert x{\vert }^{2})\) for \(x \in {\mathbb{R}}^{n}\) where f is as in (13.3.1), invoking Lemma 13.23 it follows that \(\phi \in {C}^{\infty }({\mathbb{R}}^{n})\). Finally, the condition \(\int _{{\mathbb{R}}^{n}}\phi (x)\,\mathrm{d}x = 1\) follows upon observing that based on (13.8.9) we have \(\int _{{\mathbb{R}}^{n}}{\mathrm{e}}^{ \frac{1} {\vert x{\vert }^{2}-1} }\,\mathrm{d}x = 1/C\).
Proposition 13.25.
Let \(F_{0},F_{1} \subset {\mathbb{R}}^{n}\) be two nonempty sets with the property that \(\mathrm{dist}(F_{0},F_{1}) > 0\) . Then there exists a function \(\psi: {\mathbb{R}}^{n} \rightarrow \mathbb{R}\) with the following properties:
Proof.
Let \(r:= \mathrm{dist}(F_{0},F_{1}) > 0\) and set \(\widetilde{F_{1}}:=\big\{ x \in {\mathbb{R}}^{n}:\, \mathrm{dist}\,(x,F_{1}) \leq r/4\big\}\). Also, with ϕ as in Lemma 13.24, define the function \(\theta (x):={\left (\frac{4} {r}\right)}^{n}\phi (4x/r)\) for \(x \in {\mathbb{R}}^{n}\). Then
We claim that the function \(\psi: {\mathbb{R}}^{n} \rightarrow \mathbb{R}\) defined by \(\psi:=\chi _{\widetilde{F_{1}}}{\ast}\theta\) has the desired properties. To see why this is true, note that since
from the properties of θ it is immediate that \(\psi \in {C}^{\infty }({\mathbb{R}}^{n})\) and 0 ≤ ψ ≤ 1. Furthermore, for \(\alpha \in \mathbb{N}_{0}^{n}\) and \(x \in {\mathbb{R}}^{n}\) we may write
where \(C_{\alpha }:= {4}^{\vert \alpha \vert }\int _{{\mathbb{R}}^{n}}\big\vert {\partial }^{\alpha }\phi (y)\big\vert \,\mathrm{d}y\) is a positive, finite number independent of r.
We are left with checking the fact that ψ = j on F j , j = 0, 1. First, if x ∈ F 0, then \(\vert x - y\vert \geq 3r/4\) for every \(y \in \widetilde{F_{1}}\), hence \(\theta (x - y) = 0\) for every \(y \in \widetilde{F_{1}}\), which when combined with (13.3.7) implies ψ(x) = 0. Second, if x ∈ F 1, then \(\overline{B(x,r/4)} \subseteq \widetilde{F_{1}}\). Since the support of \(\theta (x -\cdot ) \subset \overline{B(x,r/4)}\) the latter implies \(\psi (x) =\int _{{\mathbb{R}}^{n}}\theta (x - y)\,\mathrm{d}y = 1\). The proof of the proposition is complete.
A trivial yet useful consequence of the above result is as follows.
Proposition 13.26.
If \(U \subseteq {\mathbb{R}}^{n}\) is open and \(K \subset U\) is compact, then there exists a function \(\psi: {\mathbb{R}}^{n} \rightarrow \mathbb{R}\) that is of class C ∞ , satisfies 0 ≤ψ(x) ≤ 1 for every \(x \in {\mathbb{R}}^{n}\) and ψ(x) = 1 for every x ∈ K, and which has compact support, contained in U.
Proof.
Since K is a compact contained in U we have \(r:= \mathrm{dist}\,(K, {\mathbb{R}}^{n} \setminus U)\) is contained in (0, ∞). Proposition 13.25 applied with F 1: = K and \(F_{0}:=\{ x \in {\mathbb{R}}^{n}:\, \mathrm{dist}\,(x,K) > r/2\}\) then yields the desired function ψ.
13.4 Partition of Unity
Lemma 13.27.
If \(C \subset {\mathbb{R}}^{n}\) is compact, and \(U \subseteq {\mathbb{R}}^{n}\) is an open set such that \(C \subset U\) , then there exists a compact set \(D \subseteq {\mathbb{R}}^{n}\) such that \(C \subset {\mathring{D}} \subset D \subset U\) .
Proof.
Let \(V = {U}^{c} \cap \overline{B(0,R)}\), where R > 0 is large enough so that \(\overline{U} \subset B(0,R)\). Then V is compact and disjoint from C so \(r:= \mathrm{dist}(V,C) =\inf \limits _{\stackrel{x\in C}{y\in V }}\|x - y\| > 0\). Hence, if we set
then D is compact and \(C \subset {\mathring{D}} \subset D \subset U\).
Lemma 13.28.
Suppose that \(K \subseteq {\mathbb{R}}^{n}\) is a compact set and that \(\left \{O_{j}\right \}_{1\leq j\leq k}\) is a finite open cover of K. Then there are compact sets \(D_{j} \subset O_{j}\), \(1 \leq j \leq k\) , with the property that
Proof.
Set \(C_{1}:= K \setminus \bigcup _{j=2}^{k}O_{j} \subseteq O_{1}\). Since C 1 is compact, Lemma 13.27 shows that there exists a compact set D 1 with the property that \(C_{1} \subset {\mathring{D}}_{1} \subset D_{1} \subset O_{1}\). Next, proceeding inductively, suppose that \(m \in \mathbb{N}\) is such that 1 ≤ m < k and that m compact sets \(D_{1},\ldots,D_{m}\) have been constructed with the property that \(K \subset \big (\bigcup _{j=1}^{m}{\mathring{D}}_{j}\big) \cup \big (\bigcup _{j=m+1}^{k}O_{j}\big)\). Introduce
Clearly, C m + 1 is a compact subset of O m + 1. By once again invoking Lemma 13.27, there exists a compact set \(D_{m+1} \subset O_{m+1}\) such that \(C_{m+1} \subset {\mathring{D}}_{m+1}\). After k iterations, this procedure yields a family of sets C 1, …, C k which have all the desired properties.
Theorem 13.29 (Partition of unity for compact sets).
Let \(K \subset {\mathbb{R}}^{n}\) be a compact set, and let \(\{O_{j}\}_{1\leq j\leq N}\) be a finite open cover of K. Then there exists a finite collection of C ∞ functions \(\varphi _{j}: {\mathbb{R}}^{n} \rightarrow \mathbb{R}\) , 1 ≤ j ≤ N, satisfying the following properties:
-
(i)
For every 1 ≤ j ≤ N, the set \(\mathrm{supp}\,(\varphi _{j})\) is compact and contained in O j ;
-
(ii)
For every 1 ≤ j ≤ N, one has \(0 \leq \varphi _{j} \leq 1\) ;
-
(iii)
\(\sum \limits _{j=1}^{N}\varphi _{j}(x) = 1\) for every x ∈ K.
The family \(\{\varphi _{j}:\, 1 \leq j \leq N\}\) is called a partition of unity subordinate to the cover \(\{O_{j}\}_{1\leq j\leq N}\) of K.
Proof.
Let \(\left \{O_{j}\right \}_{1\leq j\leq N}\) be any finite open cover for K. From Lemma 13.28 we know that there exist compact sets \(D_{j} \subseteq O_{j}\), 1 ≤ j ≤ N, such that \(K \subset \bigcup \limits _{j=1}^{N}{\mathring{D}}_{j}\). By Proposition 13.26, for each 1 ≤ j ≤ N, choose a C ∞ function \(\eta _{j}: {\mathbb{R}}^{n} \rightarrow [0,\infty )\) that is positive on D j and has compact support in O j . It follows that \(\sum \limits _{j=1}^{N}\eta _{j}(x) > 0\) for all \(x \in \bigcup \limits _{j=1}^{N}{\mathring{D}}_{j}\), so we can define for each j ∈ { 1, …, N} the function
By Lemma 13.28, there exists a compact set \(U \subseteq {\mathbb{R}}^{n}\) with \(K \subseteq {\mathring{U}} \subseteq U \subseteq \bigcup \limits _{j=1}^{N}{\mathring{D}}_{j}\). We apply Proposition 13.26 to obtain a C ∞ function \(f: {\mathbb{R}}^{n} \rightarrow [0,1]\) that satisfies f(x) = 1 for x ∈ K and having compact support contained in \({\mathring{U}}\). Then for each j ∈ { 1, …, N} we define the function \(\varphi _{j}:= f\psi _{j}\) acting from \({\mathbb{R}}^{n}\) into \(\mathbb{R}\). It is not hard to see that each \(\varphi _{j}\) is C ∞ in \({\mathbb{R}}^{n}\), has compact support, contained in O j , \(0 \leq \varphi _{j} \leq 1\), and \(\sum \limits _{1\leq j\leq N}\varphi _{j}(x) = 1\) for all x ∈ K.
Definition 13.30.
-
(i)
A family \(\{F_{j}\}_{j\in I}\) of subsets of \({\mathbb{R}}^{n}\) is said to be locally finite in \(E \subseteq {\mathbb{R}}^{n}\) provided every x ∈ E has a neighborhood \(\mathcal{O}\subseteq {\mathbb{R}}^{n}\) with the property that the set \(\{j \in I:\, F_{j} \cap \mathcal{O}\not =\varnothing \}\) is finite.
-
(ii)
Given a collection of functions \(f_{j}: \Omega \rightarrow \mathbb{R}\) , j ∈ I, defined in some fixed subset Ω of \({\mathbb{R}}^{n}\) , the sum \(\sum \limits _{j\in I}f_{j}\) is called locally finite in \(E \subseteq {\mathbb{R}}^{n}\) provided the family of sets \(\{x \in \Omega :\, f_{j}(x)\not =0\}\), indexed by j ∈ I, is locally finite in E.
Exercise 13.31.
Show that a family {F j } j∈I of subsets of \({\mathbb{R}}^{n}\) is locally finite in the open set \(E \subseteq {\mathbb{R}}^{n}\) if and only if for every compact \(K \subseteq E\) the collection \(\{j \in I:\, F_{j} \cap K\not =\varnothing \}\) is finite.
Exercise 13.32.
Show that if the family \(\{F_{j}\}_{j\in I}\) of subsets of \({\mathbb{R}}^{n}\) is locally finite in \(E \subseteq {\mathbb{R}}^{n}\) , then \(\{\overline{F_{j}}\,\}_{j\in I}\) is also locally finite in E.
Exercise 13.33.
Show that if the family \(\{F_{j}\}_{j\in I}\) of closed subsets of \({\mathbb{R}}^{n}\) is locally finite in \(E \subseteq {\mathbb{R}}^{n}\) , then \(\bigcup \limits _{j\in I}(E \cap F_{j})\) is a relatively closed subset of E.
Proof.
Let \({x}^{{\ast}}\in E\) be such that \({x}^{{\ast}}\notin F_{j}\) for every j ∈ I. Then there exists r > 0 with the property that \(B({x}^{{\ast}},r) \cap F_{j} = \varnothing \) for every \(j \in I \setminus I_{{\ast}}\), where I ∗ is a finite subset of I. Hence, by eventually further decreasing r, it can be assumed that \(B({x}^{{\ast}},r) \cap F_{j} = \varnothing \) for every I ∗ as well. Thus, \(B({x}^{{\ast}},r) \cap E\) is a relative neighborhood of x ∗ in E that is disjoint from \(\bigcup \limits _{j\in I}F_{j}\). This proves that \(E \setminus {\Bigl (\bigcup \limits _{j\in I}F_{j}\Bigr )}\) is relatively open in E hence, \(\bigcup \limits _{j\in I}(E \cap F_{j})\) is a relatively closed subset of E.
Theorem 13.34 (Partition of unity for arbitrary open covers).
Let \(\{O_{k}\}_{k\in I}\) be an arbitrary family of open sets in \({\mathbb{R}}^{n}\) and set \(\Omega :=\bigcup \limits _{k\in I}O_{k}\) . Then there exists an at most countable collection \(\{\varphi _{j}\}_{j\in J}\) of C ∞ functions \(\varphi _{j}: \Omega \rightarrow \mathbb{R}\) satisfying the following properties:
-
(i)
For every j ∈ J there exists k ∈ I such that \(\varphi _{j}\) is compactly supported in O k;
-
(ii)
For every j ∈ J, one has \(0 \leq \varphi _{j} \leq 1\) in Ω;
-
(iii)
The family of sets \(\{x \in \Omega :\,\varphi _{j}(x)\not =0\}\) , indexed by j ∈ J, is locally finite in Ω;
-
(iv)
\(\sum \limits _{j\in J}\varphi _{j}(x) = 1\) for every x ∈ Ω.
The family \(\{\varphi _{j}\}_{j\in J}\) is called a partition of unity subordinate to the family \(\{O_{k}\}_{k\in I}\).
Proof.
Start by defining
Then \(\Omega =\bigcup \limits _{ j=1}^{\infty }\Omega _{j}\) and
Proceed now to define the compact sets
As such, from (13.4.7) and (13.4.6) we have
Finally, we define the following families of open sets
Making use of (13.4.6), for every j ≥ 3 we have that
Hence, (13.4.7), (13.4.9), and (13.4.10), imply that \(\mathcal{O}_{j}\) is an open cover for K j for every j ≥ 3. Also, from the definitions of K 2 and \(\mathcal{O}_{2}\) and (13.4.6), we obtain that \(\mathcal{O}_{2}\) is an open cover for K 2. Since the K j ’s are compact, these open covers can be refined to finite subcovers in each case. As such, for each \(j = 2,3,\ldots\), we can apply Theorem 13.29 to obtain a finite partition of unity \(\{\varphi:\,\varphi \in \Phi _{j}\}\) for K j subordinate to \(\mathcal{O}_{j}\). Also note that due to (13.4.10) and (13.4.9), we necessarily have that, for each \(j \in \{ 2,3,\ldots \}\), \(\Omega _{j} \cap O = \varnothing \) for every \(O \in \mathcal{O}_{k}\), for every \(k \in \mathbb{N}\) satisfying k ≥ j + 2. This ensures that the family \(\{\mathrm{supp}\,\varphi:\,\varphi \in \Phi _{j},\,j \geq 2\}\) is locally finite in Ω, so we can define
Given that differentiability is a local property, it follows that s is of class C ∞ in Ω. Moreover, note that s(x) > 0 for every x ∈ Ω, since \(0 \leq \varphi \leq 1\) for all \(\varphi \in \Phi _{j}\), \(j = 2,3,\ldots\), and if x ∈ Ω j , some \(j \in \{ 2,3,\ldots \}\), then \(\sum \limits _{\varphi \in \Phi _{j}}\varphi (x) = 1\). Consequently, 1 ∕ s is also a C ∞ function in Ω. It is then clear that the collection
is a partition of unity subordinate to the family of open sets \(\{O_{k}\}_{k\in I}\) (in the sense described in the statement of the theorem).
Theorem 13.35 (Partition of unity with preservation of indexes).
Let \(\{O_{k}\}_{k\in I}\) be an arbitrary family of open sets in \({\mathbb{R}}^{n}\) and set \(\Omega :=\bigcup \limits _{k\in I}O_{k}\) . Then there exists a collection \(\{\psi _{k}\}_{k\in I}\) of C ∞ functions \(\psi _{k}: \Omega \rightarrow \mathbb{R}\) satisfying the following properties:
-
(i)
For every k ∈ I the function ψ k vanishes outside of a relatively closed subset of O k ;
-
(ii)
For every k ∈ I, one has 0 ≤ψ k ≤ 1 in Ω;
-
(iii)
The family of sets \(\{x \in \Omega :\,\psi _{k}(x)\not =0\}\) , indexed by k ∈ I, is locally finite in Ω;
-
(iv)
\(\sum \limits _{k\in I}\psi _{k}(x) = 1\) for every x ∈ Ω.
Proof.
Let \(\{\varphi _{j}\}_{j\in J}\) be a partition of unity subordinate to the family \(\{O_{k}\}_{k\in I}\), and denote by f: J → I a function with the property that, for every j ∈ J, the function \(\varphi _{j}\) is compactly supported in O f(j). That this exists is guaranteed by Theorem 13.34. For every k ∈ I then define
Note that the sum is locally finite in Ω, hence \(\psi _{k}: \Omega \rightarrow \mathbb{R}\) is a well-defined nonnegative function, of class C ∞ in Ω, for every k ∈ I. In addition, the result from Exercise 13.33 shows that, for every k ∈ I, \(\varphi _{k}\) vanishes outside a relatively closed subset of O k . Furthermore, since \(\{{f}^{-1}(\{k\})\}_{k\in I}\) is a partition of J into mutually disjoint subsets, we may compute
Incidentally, this also shows that, necessarily, 0 ≤ ψ k (x) ≤ 1 for every k ∈ I and x ∈ Ω. Finally, the fact that the family of sets {x ∈ Ω: ψ k (x) ≠ 0}, k ∈ I, is locally finite in Ω is inherited from the corresponding property of the \(\varphi _{j}\)’s.
13.5 The Gamma and Beta Functions
The gamma function is defined as
It is easy to check that Γ(1) = 1, \(\Gamma (1/2) = \sqrt{\pi }\) and via integration by parts that \(\Gamma (z + 1) = z\Gamma (z)\) for \(z \in \mathbb{C}\), Re z > 0. By analytic continuation, the function Γ(z) is extended to a meromorphic function defined for all complex numbers z except for \(z = -n\), \(n \in \mathbb{N}_{0}\), where the extended function has simple poles with residue \({(-1)}^{n}/n!\) and this extension satisfies
By induction it follows that for every \(n \in \mathbb{N}\) we have
The volume of the unit ball B(0, 1) in \({\mathbb{R}}^{n}\), which we denote by | B(0, 1) | , and the surface area of the unit sphere in \({\mathbb{R}}^{n}\), denoted here by ω n − 1, have the following formulas:
Next, consider the so-called beta function
Clearly B(z, w) = B(w, z). Making the change of variables \(t = u/(u + 1)\), u ∈ (0, ∞), it follows that
whenever Re z, Re w > 0.
The basic identity relating the gamma and beta functions reads
This is easily proved starting with (13.5.8), writing \(\Gamma (z + w) =\int _{ 0}^{\infty }{t}^{z+w-1}{\mathrm{e}}^{-t}\,\mathrm{d}t\) and expressing B(z, w)Γ(z + w) as a double integral, then making the change of variables \(s:= t/(u + 1)\). A useful consequence of identity (13.5.9) is the following formula
Indeed, making the change of variables u: = (sinθ)2, the integral in the leftmost side of (13.5.10) becomes \(\frac{1} {2}\int _{0}^{1}{u}^{(a-1)/2}{(1 - u)}^{(b-1)/2}\,\mathrm{d}u\).
For further reference, let us also note here that
whenever a, b > − 1. This is proved by splitting the domain of integration into \((0,\pi /2) \cup (\pi /2,\pi )\), making a change of variables \(\theta \mapsto \theta -\pi /2\) in the second integral, and invoking (13.5.10).
13.6 Surfaces in \({\mathbb{R}}^{n}\) and Surface Integrals
Definition 13.36.
Given n ≥ 2 and \(k \in \mathbb{N} \cup \{\infty \}\) , a C k surface (or, surface of class C k) in \({\mathbb{R}}^{n}\) is a subset Σ of \({\mathbb{R}}^{n}\) with the property that for every x ∗ ∈ Σ there exists an open neighborhood U(x ∗) such that
where \(\mathcal{O}\) is an open subset of \({\mathbb{R}}^{n-1}\) and
where DP is the Jacobian matrix of P, that is,
The function P in (13.6.2) is called a local parametrization of class C k near x∗ and \(\Sigma \cap U({x}^{{\ast}})\) a parametrizable patch.
In the case when (13.6.1) holds when we formally take \(r = +\infty \) , that is, in the case when \(\Sigma = P(\mathcal{O})\) , we call P a global parametrization of the surface Σ.
Proposition 13.37.
If \(\mathcal{O}\) is an open subset of \({\mathbb{R}}^{n-1}\) and \(P: \mathcal{O}\rightarrow {\mathbb{R}}^{n}\) satisfies (13.6.2)–(13.6.3) for k = 1, then \(P: \mathcal{O}\rightarrow P(\mathcal{O})\) is a homeomorphism.
Definition 13.38.
Assume n ≥ 3. If \(v_{1} = (v_{11},\ldots,v_{1n}),\ldots,v_{n-1} = (v_{n-1\;1},\ldots,v_{n-1\,n})\) are n − 1 vectors in \({\mathbb{R}}^{n}\) , their cross product is defined as
where the determinant is understood as computed by formally expanding it with respect to the last row, the result being a vector in \({\mathbb{R}}^{n}\) . More precisely,
Definition 13.39.
Let \(\Sigma \subset {\mathbb{R}}^{n}\) , n ≥ 2, be a C 1 surface and assume that \(P: \mathcal{O}\rightarrow {\mathbb{R}}^{n}\) , with \(\mathcal{O}\) open subset of \({\mathbb{R}}^{n-1}\) , is a local parametrization of Σ of class C 1 near some point X ∗ ∈ Σ. Also, suppose that \(f: \Sigma \rightarrow \mathbb{R}\) is a continuous function on Σ that vanishes outside of a compact subset of \(P(\mathcal{O})\) . We then define
if n ≥ 3, and
In (13.6.5), dσ stands for the surface measure (or, surface area element), whereas in (13.6.6), dσ stands for the arc-length measure.
13.7 Integration by Parts and Green’s Formula
Definition 13.40.
We say that a nonempty open set \(\Omega \subseteq {\mathbb{R}}^{n}\) , where n ≥ 2, is a C k domain (or, a domain of class C k ), for some \(k \in \mathbb{N}_{0} \cup \{\infty \}\) , provided the following holds. For every point \({x}^{{\ast}}\in \partial \Omega \) there exist R > 0, an open interval \(I \subset \mathbb{R}\) with 0 ∈ I, a rigid transformation \(T: {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}\) with T(x ∗ ) = 0, along with a function ϕ of class C k that maps \(B(0,R) \subseteq {\mathbb{R}}^{n-1}\) into I with the property that ϕ(0) = 0 and, if \(\mathcal{C}\) denotes the (open) cylinder \(B(0,R) \times I \subseteq {\mathbb{R}}^{n-1} \times \mathbb{R} = {\mathbb{R}}^{n}\) , then
If \(\Omega \subseteq {\mathbb{R}}^{n}\) is a C k domain for some \(k \in \mathbb{N}_{0} \cup \{\infty \}\), then it may be easily verified that \(\partial \Omega \) is a C k surface.
Theorem 13.41 (Integration by parts formula).
Suppose \(\Omega \subset {\mathbb{R}}^{n}\) is a domain of class C 1 and \(\nu = (\nu _{1},\ldots,\nu _{n})\) denotes its outward unit normal. Let \(k \in \{ 1,\ldots,n\}\) and assume \(f,g \in {C}^{1}(\Omega ) \cap {C}^{0}(\overline{\Omega }\,)\) are such that \(\partial _{k}f\), \(\partial _{k}g \in {L}^{1}(\Omega )\) and there exists a compact set \(K \subset {\mathbb{R}}^{n}\) with the property that f = 0 on \(\Omega \setminus K\) . Then,
where σ is the surface measure on \(\partial \Omega \) .
For the sense in which the last integral in (13.7.4) should be understood see Definition 13.39.
An immediate corollary of Theorem 13.41 is Green’s formula that is stated next (recall (7.1.14)).
Theorem 13.42 (Green’s formula).
Suppose \(\Omega \subset {\mathbb{R}}^{n}\) is a bounded domain of class C 1 and ν denotes its outward unit normal. If \(f,g \in {C}^{2}(\overline{\Omega })\) , then
13.8 Polar Coordinates and Integrals on Spheres
Assume that n ≥ 3 and R > 0 are fixed. For ρ ∈ (0, R), θ j ∈ (0, π), \(j \in \{ 1,\ldots,n - 2\}\), and θ n − 1 ∈ (0, 2π), set
The variables \(\theta _{1},\ldots,\theta _{n-1},\rho\) are called polar coordinates.
Definition 13.43.
Assume that \(x_{{\ast}}\in {\mathbb{R}}^{n}\) , n ≥ 3, is a fixed point, and R > 0 is arbitrary. The standard parametrization of the ball B(x ∗,R) is defined as
where \(x_{1},\ldots,x_{n}\) are as in (13.8.1).
The function \(\mathcal{P}\) in (13.8.2) is injective, of class C ∞, takes values in B(x ∗ , R), its image differs from B(x ∗ , R) by a subset of measure zero and
at every point in its domain, where \(D\mathcal{P}\) denotes the Jacobian of \(\mathcal{P}\). Using this standard parametrization for the unit sphere in \({\mathbb{R}}^{n}\), we see that
This parametrization of the sphere B(x ∗ , R) may also be used to prove the following theorem.
Theorem 13.44 (Spherical Fubini and polar coordinates).
Let \(f \in L_{\mathrm{loc}}^{1}({\mathbb{R}}^{n})\) , n ≥ 2. Then for each \(x_{{\ast}}\in {\mathbb{R}}^{n}\) and each R > 0 the following formulas hold:
Moreover, if \(f \in {L}^{1}({\mathbb{R}}^{n})\) , then
Note that if \(P: \mathcal{O}\rightarrow {S}^{n-1}\) is a parametrization of the unit sphere in \({\mathbb{R}}^{n}\) and if \(\mathcal{R}\) is a unitary transformation in \({\mathbb{R}}^{n}\), then \(\mathcal{R}\circ P: \mathcal{O}\rightarrow {S}^{n-1}\) is also a parametrization of the unit sphere in \({\mathbb{R}}^{n}\). Indeed, this function is injective, has C 1 components, its image is S n − 1 (up to a negligible set) and
Hence,
The same type of reasoning also yields the following result.
Proposition 13.45.
For each \(x = (x_{1},\ldots,x_{n}) \in {\mathbb{R}}^{n}\) , define
Then
Proposition 13.46.
Let \(v \in {\mathbb{R}}^{n} \setminus \{ 0\}\) , n ≥ 2, be fixed. Then for any real-valued function f defined on the real line, there holds
Proof.
Since integrals over the unit sphere are invariant under orthogonal transformations, we may assume that \(v/\vert v\vert = e_{1}\) and, hence, using polar coordinates and (13.8.3), we have
Making the change of variables \(s:=\cos \,\varphi _{1}\) in the last integral above shows that this matches the right-hand side of (13.8.15).
Proposition 13.47.
Let \(f \in {C}^{0}(\mathbb{R})\) be positive homogeneous of degree \(m \in \mathbb{R}\) and fix \(\eta = (\eta _{1},\ldots,\eta _{n}) \in {\mathbb{R}}^{n}\) . Then if \(n \in \mathbb{N}\) with n ≥ 2, for \(j,k \in \{ 1,\ldots,n\}\) one has
where
Proof.
For \(j,k \in \{ 1,\ldots,n\}\) set
and define the quadratic form
Observe that we can write \(Q(\zeta,\eta ) =\int _{{S}^{n-1}}f(\eta \cdot \xi ){(\zeta \cdot \xi )}^{2}\,\mathrm{d}\sigma (\xi )\). By the invariance under rotations of integrals over S n − 1 (see (13.8.11)), we have that for any rotation R in \({\mathbb{R}}^{n}\)
A direct computation also gives that
Next we claim that
To show that (13.8.23) holds, we first observe that it suffices to prove (13.8.23) when η = e 1. Indeed, if we assume that (13.8.23) is true for η = e 1, then for an arbitrary η let R be the rotation such that R η = e 1. Then if we also take into account (13.8.20), we have
Hence, (13.8.23) will follow if we prove that
To see the later we let \(A_{jk}:=\int _{{S}^{n-1}}f(\xi _{1})\xi _{j}\xi _{k}\,\mathrm{d}\sigma (\xi )\) for \(j,k \in \{ 1,\ldots,n\}\). Assume j ≠ k. Then either j ≠ 1 or k ≠ 1. If, say, j ≠ 1 we use (13.8.13) to conclude that A jk = 0 in this case. A similar reasoning applies if k ≠ 1. Clearly, \(A_{11} =\int _{{S}^{n-1}}f(\xi _{1})\xi _{1}^{2}\,\mathrm{d}\sigma (\xi )\). As for the case \(j = k\not =1\), we first observe that \(A_{22} =\ldots = A_{nn}\) since A jj is independent of \(j \in \{ 2,\ldots,n\}\) due to (13.8.14). Moreover, \(\sum \limits _{j=1}^{n}A_{jj} =\int _{{S}^{n-1}}f(\xi _{1})\,\mathrm{d}\sigma (\xi )\), which in turn implies that \(A_{jj} = \frac{1} {n-1}{\Bigl (\int _{{S}^{n-1}}f(\xi _{1})\,\mathrm{d}\sigma (\xi ) - A_{11}\Bigr )} =\alpha\) for each \(j = 2,\ldots,n\). Combining all these we have that for ζ ∈ S n − 1,
This concludes the proof of (13.8.25) which, in turn, implies (13.8.23). Now if ζ, \(\eta \in {\mathbb{R}}^{n} \setminus \{ 0\}\), we make use of (13.8.22) and (13.8.23) to write
which proves (13.8.17).
Proposition 13.48.
Consider f(t):= |t| for \(t \in \mathbb{R}\) , and let α,β be as in (13.8.17) for this choice of f. Then
where ω n−2 denotes the surface measure of the unit ball in \({\mathbb{R}}^{n-1}\) .
Proof.
Using the standard parametrization of S n − 1 (see (13.8.1) with R = 1) we have
The change of variables \(\theta \!=\!\pi -y\) yields \(-\int _{\frac{\pi }{ 2} }^{\pi }\cos \theta {(\sin \theta )}^{n}\,\mathrm{d}\theta \! =\!\int _{ 0}^{ \frac{\pi }{ 2} }\cos y{(\sin y)}^{n}\,\mathrm{d}y\). Using this back in (13.8.29) then gives
Similar arguments in computing β give that
This finishes the proof.
Recall from 0.0.7 that \({z}^{\alpha } = z_{1}^{\alpha _{1}}z_{2}^{\alpha _{2}}\cdots z_{n}^{\alpha _{n}}\) whenever \(z = (z_{1},\ldots,z_{n}) \in {\mathbb{R}}^{n}\) and \(\alpha = (\alpha _{1},\alpha _{2},\ldots,\alpha _{n}) \in \mathbb{N}_{0}^{n}\). Let us also introduce
The next proposition deals with the issue of integrating arbitrary monomials on the unit sphere centered at the origin.
Proposition 13.49.
For each multi-index \(\alpha = (\alpha _{1},\ldots,\alpha _{n}) \in \mathbb{N}_{0}^{n}\),
where Γ is the gamma function introduced in (13.5.1).
Proof.
Fix an arbitrary \(k \in \mathbb{N}\) and set
Also, with a “dot” standing for the standard inner product in \({\mathbb{R}}^{n}\), introduce
Then (13.2.1) implies that
Let us also observe here that, if x ∈ S n − 1 is arbitrary but fixed and if \(\mathcal{R}\) is a rotation about the origin in \({\mathbb{R}}^{n}\) such that \({\mathcal{R}}^{-1}x = e_{n}:= (0,\ldots,0,1) \in {\mathbb{R}}^{n}\), then by (13.8.35) and the rotation invariance of integrals on S n − 1 (cf. (13.8.11)), we have
By the homogeneity of Q k , (13.8.37) implies that \(Q_{k}(x) = \vert x{\vert }^{k}Q_{k}(e_{n})\) for all \(x \in {\mathbb{R}}^{n}\) and, hence,
We now consider two cases:
Case I: k is odd. In this scenario, the mapping \({S}^{n-1} \ni z\mapsto z_{n}^{k} \in \mathbb{R}\) is odd. In particular, \(Q_{k}(x) =\int _{{S}^{n-1}}z_{n}^{k}\,\mathrm{d}\sigma (z) = 0\). This, in turn, along with (13.8.38) then force \(\sum \limits _{\vert \alpha \vert =k}\frac{k!} {\alpha !} \,q_{\alpha }{x}^{\alpha } = 0\) for every \(x \in {\mathbb{R}}^{n}\). From (13.2.5) it is easy to deduce that for each \(\beta \in \mathbb{N}_{0}^{n}\) we have
We may therefore conclude that \(q_{\gamma } = \partial _{x}^{\gamma }{\Bigl [\sum _{\vert \alpha \vert =k}\frac{1} {\alpha !} \,q_{\alpha }{x}^{\alpha }\Bigr ]} = 0\) for each \(\gamma \in \mathbb{N}_{0}^{n}\) with | γ | = k, in agreement with (13.8.33).
Case II: k is even. Suppose k = 2m for some \(m \in \mathbb{N}\). Then, \(\vert x{\vert }^{k} =\sum \limits _{\vert \beta \vert =m}\frac{m!} {\beta !} {x}^{2\beta }\) and (13.8.38) becomes
Fix \(\gamma \in \mathbb{N}_{0}^{n}\) such that | γ | = k and observe that
This, (13.8.40), and (13.8.39) then imply that
We are now left with computing Q 2m (e n ) when \(m \in \mathbb{N}\). Using spherical coordinates, a direct computation gives that
by (13.8.4) and (13.5.6) (considered with n − 1 in place of n), and (13.5.11). This once again agrees with (13.8.33), and the proof of Proposition 13.49 is finished.
A simple useful consequence of the general formula (13.8.33) is the fact that
13.9 Tables of Fourier Transforms
Fourier Transforms of Schwartz Functions
Fourier Transforms of Tempered Distributions in \(\mathbb{R}\)Below \(x,\xi \in \mathbb{R}\).
Fourier Transforms of Tempered DistributionsBelow \(x,\xi \in {\mathbb{R}}^{n}\) and \(x^{\prime},\xi \prime \in {\mathbb{R}}^{n-1}\).
References
H. L. Royden, Real Analysis, Macmillian Publishing Co., Inc., New York, 1968.
W. Rudin, Real and Complex Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., 1986.
W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., 1991.
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Mitrea, D. (2013). Appendix. In: Distributions, Partial Differential Equations, and Harmonic Analysis. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8208-6_13
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