Hilbert Spaces

Abstract

Arguably, Hilbert spaces and operators on those spaces are the most popular objects of study in functional analysis because they have been very useful in many applications.

Appendices

Notes

The inner product generalizes the dot product—a concept well known from Euclidean geometry and physics. In the Euclidean space $\mathbf R^3$, the dot product of two vectors $\varvec{x}=(x_1,x_2,x_3)$ and $\varvec{y}=(y_1,y_2,y_3)$ is defined by

$$ \varvec{x}\cdot \varvec{y}=x_1y_1+x_2y_2+x_3y_3, $$

and equivalently, by $\varvec{x}\cdot \varvec{y}=|\varvec{x}||\varvec{y}|\cos \vartheta$, where $|\varvec{x}|$ and $|\varvec{y}|$ are the respective magnitudes of the vectors $\varvec{x}$ and $\varvec{y}$, and $\vartheta$ is the angle between these two vectors. In physics, for instance, the mechanical work is the dot product of the force and displacement vectors.

Theorem 7.7 is an instance of a more general result that can be easily deduced from the proof of the theorem (cf. Exercise 7.23). Namely, let E be a closed convex subset of a Hilbert space. Then for every $x\in H$, there is a unique $y\in E$ such that

$$ \Vert x-y\Vert =d(x, E)=\inf _{u\in E}\Vert x-u\Vert . $$

According to Lemma 7.4, a family $\{a_i\}_{i\in J}$ of nonnegative numbers is summable in $\mathbf R$ only if the set of indices i such that $a_i\ne 0$ is at most countable. Thus, the study of summable families reduces essentially to the study of summable sequences. However, as examples in Section 7.3 and Appendix A demonstrate, we have to consider possibly uncountable families in the main theorems of Section 7.3. Our definition of summable families (cf. Definition 7.6) is adopted from Bourbaki (1966), Section 5.1.

All bases of a Hilbert space H have the same cardinality. This result can be easily established for separable spaces (including finite-dimensional spaces). However, for a general H this is not a simple fact. The proof can be found, for instance, in Dunford and Schwartz (1958), (Chapter IV, Section 4, Theorem 14). The cardinality of a basis of the space H is called the Hilbert dimension of H. It can be proved (cf. Corollary 7.3) that every two Hilbert spaces of the same dimension are isomorphic.

We did not prove that the space $\ell _2(J)$ from Example 7.13 is indeed a Hilbert space. For this, the reader is referred to Appendix A.2.

The vector space X from Example 7.14 becomes a normed space if the norm is defined by

$$ \Vert f\Vert =\sup _{x\in \mathbf R}|f(x)|. $$

The completion of X with respect to this norm is a normed space of almost periodic functions on $\mathbf R$ (Dunford and Schwartz 1958, Chapter IV, Section 7). Note that this norm is not the same as the norm generated by the inner product introduced in Example 7.14.

There are unbounded operators T on a Hilbert space in quantum mechanics that satisfy the relation $\langle Tx,y\rangle =\langle x, Ty\rangle$. By the Hellinger–Toeplitz theorem (cf. Corollary 7.4), such operators cannot be defined on the entire space. The theory of unbounded operators is beyond the scope of this book.

The classes of normal, self-adjoint, unitary, and projection operators have been studied extensively because of their role in more advanced parts of functional analysis and its applications (see, for instance, Chapter VI in Dunford and Schwartz 1958), where other important classes of operators are also introduced and studied).

The conditions $P^2=P=P^*$ are often used as the definition of the projection operator P. Projection operators play a significant role in the spectral theory of self-adjoint operators.

In geometry, Apollonius’s identity (cf. Exercise 7.10) states that the sum of the squares of any two sides of any triangle equals twice the square of half the third side, together with twice the square of the median bisecting the third side.

Exercises

7.1.

Let X be an inner product space. Show that

• 1. $\langle \lambda x+\mu y,z\rangle =\lambda \langle x,z\rangle +\mu \langle y, z\rangle$,

• 2. $\langle x,\lambda y+\mu z\rangle =\overline{\lambda }\langle x,y\rangle +\overline{\mu }\langle x, z\rangle$,

for every $x,y, z\in X$ and $\lambda ,\mu \in \mathbf F$.

7.2.

Prove that (7.1) defines an inner product on $\ell _2$.

7.3.

Prove that the integral of a continuous nonnegative function x(t) on the interval [a, b] is zero if and only if the function x is the zero function.

7.4.

Let $X_1$ be the vector space C[a, b] endowed with the norm $\Vert \cdot \Vert _\infty$, and $X_2$ the same vector space endowed with the norm

$$ \Vert x\Vert =\sqrt{\int _a^b|x(t)|^2\, dt} $$

(cf. Example 7.2). Show that the identity map $x\rightarrow x$ of $X_1$ onto $X_2$ is continuous.

7.5.

Show that $\langle x, 0\rangle =\langle 0,y\rangle =0$ for all x and y in an inner product space.

7.6.

Show that $\Vert \lambda x\Vert =|\lambda |\Vert x\Vert$ for every $\lambda \in \mathbf F$ and $x\in X$, where X is an inner product space.

7.7.

Show that $\langle x, y\rangle$ is an inner product on the space H in the proof of Theorem 7.5.

7.8.

Let Y be a subspace of a Hilbert space H. Prove that

• (a) Y is complete if and only if it is closed in H.

• (b) If Y is finite-dimensional, then Y is complete.

7.9.

Let x and y be vectors in an inner product space X. Show that

$$ \langle x,x\rangle \langle y, y\rangle \ge \genfrac{}{}{0.4pt}{}{1}{4}\big (\langle x,y\rangle +\langle y, x\rangle \big )^2. $$

7.10.

(Apollonius’s identity.) Let x, y, and z be vectors in an inner product space X. Show that

$$ \Vert z-x\Vert ^2+\Vert z-y\Vert ^2=\tfrac{1}{2}\Vert x-y\Vert ^2+2\Vert z-\tfrac{1}{2}(x+y)\Vert ^2. $$

7.11.

Let $(x_n)$ be a sequence in an inner product space. Show that the conditions $\Vert x_n\Vert \rightarrow \Vert x\Vert$ and $\langle x_n,x\rangle \rightarrow \langle x, x\rangle$ imply $x_n\rightarrow x$.

7.12.

Show that

$$ X=\Big \{x=(x_n)\in \ell _2:\sum _{n=1}^\infty x_n\slash n=0\Big \} $$

is a closed subspace of $\ell _2$.

7.13.

Let x and y be linearly independent vectors in an inner product space such that $\Vert x\Vert =\Vert y\Vert =1$. Show that

$$ \Vert tx+(1-t)y\Vert<1,\qquad \text {for}\; 0<t<1. $$

7.14.

If a vector x is orthogonal to every vector in the set $\{x_1,\ldots , x_n\}$, then it is orthogonal to every linear combination of these vectors.

7.15.

Prove that in an inner product space, $x\,\bot \, y$ if and only if

$$ \Vert x+\lambda y\Vert =\Vert x-\lambda y\Vert , $$

for all scalars $\lambda \in \mathbf F$.

7.16.

Show that in 1-dimensional Hilbert space, $x\,\bot \, y$ if and only if at least one of the vectors x and y is the zero vector.

7.17.

Let X be an inner product space and E a nonempty subset of X. Show that

$$ E^\bot =\bigcap _{x\in E}\{x\}^\bot . $$

7.18.

• (a) Prove that for every two subspaces $X_1$ and $X_2$ of a Hilbert space,

  $$ (X_1+X_2)^\bot =X_1^\bot \cap L_2^\bot . $$

• (b) Prove that for every two closed subspaces $X_1$ and $X_2$ of a Hilbert space,

  $$ (X_1\cap X_2)^\bot =\overline{X_1^\bot + X_2^\bot }. $$

7.19.

Let X be an inner product space and E a nonempty subset of X. Prove the following properties:

• (a) If $x\,\bot \, E$, then $x\,\bot \,\overline{\text {span} E}$.

• (b) If E is dense in X and $x\,\bot \, E$, then $x=0$.

• (c) The set $E^\bot$ is a closed subspace of X.

7.20.

Show that the space X in Example 7.8 is not complete and the subspace E is closed in X. Also show that $E\oplus E^\bot =\{x\in X:x(0)=0\}$.

7.21.

(Pythagorean theorem.) Let X be an inner product space. Prove that

• (a) If $x\,\bot \, y$ in X, then $\Vert x+y\Vert ^2=\Vert x\Vert ^2+\Vert y\Vert ^2$.

• (b) If $\{x_1,\ldots , x_n\}$ is a set of mutually orthogonal vectors in X, then

  $$ \bigg \Vert \sum _{k=1}^n x_k\bigg \Vert ^2=\sum _{k=1}^n\Vert x_k\Vert ^2. $$

• (c) If $(x_n)$ is a sequence of mutually orthogonal vectors in X such that the sum $\sum _{k=1}^\infty x_k$ converges, then the sum $\sum _{k=1}^\infty \Vert x_k\Vert ^2$ also converges and

  $$ \bigg \Vert \sum _{k=1}^\infty x_k\bigg \Vert ^2=\sum _{k=1}^\infty \Vert x_k\Vert ^2. $$

7.22.

• (a) Let $E_1$ and $E_2$ be subspaces of an inner product space. Prove that $E_1\,\bot \, E_2$ if and only if

  $$ \Vert x_1+x_2\Vert ^2=\Vert x_1\Vert ^2+\Vert x_2\Vert ^2 $$

  whenever $x_1\in E_1$, $x_2\in E_2$.

• (b) In contrast to part (a), give an example of a Hilbert space H and vectors $x_1,x_2\in H$ such that $\Vert x_1+x_2\Vert ^2=\Vert x_1\Vert ^2+\Vert x_2\Vert ^2$, but $\langle x_1,x_2\rangle \ne 0$.

7.23.

Let E be a closed convex subset of a Hilbert space. Prove that for every $x\in H$, there is a unique $y\in E$ such that

$$ \Vert x-y\Vert =d(x, E)=\inf _{u\in E}\Vert x-u\Vert . $$

Show that the statement does not necessarily hold if E is not closed or not convex.

7.24.

Prove that the family

$$ \Big \{\genfrac{}{}{0.4pt}{}{1}{\sqrt{\pi }}\sin (nt),\quad \genfrac{}{}{0.4pt}{}{1}{\sqrt{\pi }}\cos (nt),\quad \genfrac{}{}{0.4pt}{}{1}{\sqrt{2\pi }}\Big \}_{n\in \mathbf N} $$

from Example 7.14 is orthonormal.

7.25.

Prove Lemma 7.3.

7.26.

Let $(e_k)$ be an orthonormal sequence in an inner product space X. Prove that

$$ \sum _{k=1}^\infty |\langle x,e_k\rangle \langle y, e_k\rangle |\le \Vert x\Vert \Vert y\Vert , $$

for all $x, y\in X$.

7.27.

Let $\{x_i\}_{i\in J}$ be a summable family of nonnegative numbers with sum x. Prove that for every nonempty subset $J_0$ of J, one has $\sum _{i\in J_0}x_i\le x$.

7.28.

Let J be a countable set and $\{a_i\}_{i\in J}$ a summable family of real or complex numbers with sum a. Prove that for every enumeration $J=\{i_1,i_2,\ldots \}$, one has

$$ \sum _{k=1}^\infty a_{i_k}=a. $$

7.29.

Show that the family $\{e^{\varvec{i}\lambda t}\}_{\lambda \in \mathbf R}$ in Example 7.14 is orthonormal and total in the space X.

7.30.

Let P be the vector space of all real polynomials on $[-1,1]$. Show that

$$ \langle x, y\rangle =\int _{-1}^1x(t)y(t)\, dt $$

defines an inner product on P. Use the Gram–Schmidt process to orthonormalize the set $\{1,t, t^2\}$.

7.31.

Show that a countable subset S of a vector space V contains a linearly independent subset L such that $\text {span}\,L=\text {span}\, S$.

7.32.

If $\langle x_1,y\rangle =\langle x_2,y\rangle$ for all y in an inner product space X, then $x_1=x_2$.

7.33.

Show that (7.17) defines an inner product on $H^*$ that induces a norm on $H^*$.

7.34.

Let S be a bounded sesquilinear form on $X\times Y$. Show that

$$ \Vert S\Vert =\sup \bigg \{\genfrac{}{}{0.4pt}{}{|S(x, y)|}{\Vert x\Vert \Vert y\Vert }:x\in X\setminus \{0\},\; y\in Y\setminus \{0\}\bigg \} $$

and

$$ |S(x, y)|\le \Vert S\Vert \Vert x\Vert \Vert y\Vert , $$

for all $x\in X$ and $y\in Y$.

7.35.

Let X and Y be inner product spaces and $T:X\rightarrow Y$ a bounded linear operator. Show that $T=0$ if and only if $\langle Tx, y\rangle =0$ for all $x\in X$ and $y\in Y$.

7.36.

Let X be a complex inner product space and $T:X\rightarrow X$ a bounded linear operator. Show that $\langle Tx, x\rangle =0$ for all $x\in X$ implies $T=0$.

Show that the conclusion may not hold if X is real. (Hint: Consider rotations in the real plane $\mathbf R^2$.)

7.37.

Let $T:\ell _2\rightarrow \ell _2$ be defined by

$$ T:(x_1,\ldots , x_n,\ldots )\mapsto (x_1,\ldots ,\tfrac{1}{n}x_n,\ldots ). $$

Show that $\mathcal{R}(T)$ is not closed in $\ell _2$.

7.38.

Show that $\overline{\mathcal{R}(T^*)}=\mathcal{N}(T)^\bot$ and $\overline{\mathcal{R}(T)}=\mathcal{N}(T^*)^\bot$ (cf. Theorem 7.23).

7.39.

Let $S=I+T^*T$, where $T:H\rightarrow H$ is a bounded operator. Show that S is a one-to-one mapping from H to S(H).

7.40.

Let T be a bijective self-adjoint operator on a Hilbert space. Show that $T^{-1}$ is self-adjoint.

7.41.

Verify the statements in Example 7.19.

7.42.

Show that $T^*-\overline{\lambda }I$ is the adjoint operator of $T-\lambda I$, where T is a bounded operator on a Hilbert space and $\lambda \in \mathbf F$.

7.43.

Show that the operator $T-\lambda I$, where T is a normal operator on a Hilbert space, is also normal (cf. Exercise 7.42).

7.44.

Show that the right shift operator on $\ell _2$ has no eigenvalues, whereas every $\lambda$ such that $|\lambda |<1$ is an eigenvalue of the left shift operator on $\ell _2$ (cf. Example 7.16).

7.45.

Show that if T is a self-adjoint operator, then so is $T^n$, where $n\in \mathbf N$.

7.46.

Let T be a bounded operator on a complex Hilbert space H.

• (a) Show that the operators

  $$ T_1=\genfrac{}{}{0.4pt}{}{1}{2}(T+T^*)\qquad \text {and}\qquad T_1=\genfrac{}{}{0.4pt}{}{1}{2\varvec{i}}(T-T^*) $$

  are self-adjoint.

• (b) Show that T is normal if and only if the operators $T_1$ and $T_2$ commute.

7.47.

Prove that if $T:H\rightarrow H$ is a self-adjoint operator and $T\ne 0$, then $T^n\ne 0$ for all $n\in \mathbf N$.

7.48.

Show that an isometric operator $T:H\rightarrow H$ that is not unitary maps the Hilbert space H onto a proper closed subspace of H.

7.49.

Let X be a finite-dimensional inner product space and $T:X\rightarrow X$ an isometric linear operator. Show that T is unitary.

7.50.

Suppose that a sequence $(T_n)$ of normal operators on a Hilbert space H converges (in operator norm) to an operator T. Show that T is a normal operator.

7.51.

Suppose that S and T are normal operators such that $ST^*=T^*S$. Show that $S+T$ and ST are normal operators.

7.52.

Show that if P is a projection operator, then $I-P$ is also a projection operator. Moreover, $\Vert I-P\Vert =1$ if $P\ne I$.

7.53.

If P is a projection operator on a Hilbert space H, then

$$ \Vert Px-Py\Vert \le \Vert x-y\Vert , $$

for all $x, y\in H$, that is, P is a nonexpansive operator.

7.54.

Show that two closed subspaces E and $E'$ of a Hilbert space are orthogonal if and only if $P_EP_{E'}=0$.

7.55.

If P and Q are projections of a Hilbert space H onto closed subspaces E and $E'$, respectively, and $PQ=QP$, then

$$ P+Q-PQ $$

is a projection of H onto $E+E'$.

7.56.

Prove that $P_EP_{E'}=P_E$ if and only if $E\subseteq E'$.

7.57.

Show that U is a self-adjoint unitary operator if and only if $U=2P-I$ for some projection operator P.

7.58.

Show that U is a self-adjoint unitary operator on a Hilbert space H if and only if there exist orthogonal closed subspaces $E_1$, $E_2$ such that H is the direct sum $E_1\oplus E_2$ and for every $x=x_1+x_2$ with $x_1\in E_1$, $x_2\in E_2$,

$$ Ux=x_1-x_2, $$

that is, U is a reflection.