Normed Spaces

Abstract

The first two sections of this chapter are concerned with basic properties of normed spaces and linear operators on them. The concept of a Schauder basis is central in Section 4.1. These bases are useful tools in investigating sequence spaces and linear functionals on them. Section 4.2 is concerned mostly with bounded operators and the space of bounded operators from a normed space into another normed space. The central result of this section establishes the equivalence of continuity and boundedness of a linear operator. Although the concept of the dual space is introduced in Section 4.2, we postpone studies of linear functionals and duality until Chapter 5. Several examples of linear operators are also presented in Section 4.2.

Notes

As they are in real and complex analysis, convergent and absolutely convergent series are important tools in functional analysis. For instance, in this book, we use series to prove the famous open mapping theorem in Chapter 6.

There are critical differences between the concepts of Hamel and Schauder basis. An obvious one is clearly that a Schauder basis is a sequence of vectors, whereas a Hamel basis is just a set of vectors in the space.

According to Theorem 4.1, a normed space with a Schauder basis is separable. However, as was shown in 1973 by Per Enflo (Acta Mathematica 130, pp. 309–317), there are separable spaces without Schauder basis. Of course, these spaces have Hamel bases.

The operator of indefinite integration introduced in Example 4.6 on the space C[0, 1] can be defined by the same formula on the normed space $L_2[0,1]$ (the space of Lebesgue measurable functions x on [0, 1] such that the Lebesgue integral $\int _0^1|x(t)|^2\, dt$ is bounded and with the norm $\Vert x\Vert =\sqrt{\int _0^1|x(t)|^2\, dt}$). In this case, this operator is known as the Volterra operator. The norm of the Volterra operator is $2\slash \pi$ (cf. Example 7.3 in Chapter 7 and Problem 148 in Halmos 1967).

By Theorem 4.12, every two n-dimensional normed spaces are isomorphic. In fact, every two n-dimensional metric vector spaces are isomorphic, that is, they are isomorphic as vector spaces and homeomorphic as metric spaces. Specifically, all spaces $\ell _p^n$, $p\in (0,\infty ]$ (cf. Section 3.3), are isomorphic (cf. Exercise 4.17). Moreover, every two Hausdorff topological vector spaces of the same finite dimension are isomorphic (Rudin 1973, Theorem 1.21).

The multiplication operators from Example 4.7 are, in a natural sense, the canonical examples of normal operators. This is established in the spectral theory in Hilbert spaces, which is beyond the scope of this book.

Exercises

4.1.

Give an example of an absolutely convergent series in a normed space X that is not convergent. (Hint: Try $X=c_{00}$.)

4.2.

(a) If in a normed space X, the absolute convergence of every series always implies convergence of the series, show that X is a Banach space.

(b) Show that in a Banach space, an absolutely convergent series is convergent.

4.3.

Show that vectors $(e_n)$, where $e_n$ is the sequence whose nth term is 1 and all other terms are zero,

$$\begin{aligned} e_1&=(1,0,0,\ldots ),\\ e_2&=(0,1,0,\ldots ),\\&\cdots \end{aligned}$$

form a Schauder basis in $\ell _p$ for every $p\in [1,\infty )$, and in the spaces $c_0$ and $c_{00}$.

4.4.

Prove that a mapping from a metric space X to a metric space Y is continuous if and only if the inverse image of every closed subset of Y is closed in X (cf. Theorem 2.11).

4.5.

Show that a linear operator is bounded if and only if it maps bounded sets in X into bounded subsets of Y.

4.6.

Prove Theorem 4.5.

4.7.

Show that the integral operator T in Example 4.3 is linear and maps C[0, 1] to C[0, 1].

4.8.

Let f and g be two bounded real-valued functions on the set $A\subseteq \mathbf R$. Show that

$$ \sup _{x\in A}(f(x)+g(x))\le \sup _{x\in A}f(x)+\sup _{x\in A}g(x). $$

4.9.

Prove that operators of the left and right shift on $\ell _p$ (cf. Example 4.4) are bounded and $\Vert T_l\Vert =\Vert T_r\Vert =1$.

4.10.

(Bounded linear extension theorem) Let X be a normed space and $\widetilde{X}$ its Banach completion. If f is a bounded linear functional on X, then there exists a unique linear functional $\widetilde{f}$ on $\widetilde{X}$ such that $\widetilde{f}|_X=f$ and $\Vert \widetilde{f}\Vert =\Vert f\Vert$.

4.11.

Let X, Y, and Z be normed spaces and $S:X\rightarrow Y$ and $T:Y\rightarrow Z$ bounded operators. Prove that

$$ \Vert TS\Vert \le \Vert T\Vert \Vert S\Vert . $$

4.12.

Let T be a bounded operator from a normed space X to a normed space Y. Prove that for every $x\in X$ and $r>0$,

$$ \sup _{y\in B(x,r)}\Vert Ty\Vert \ge \Vert T\Vert \, r. $$

4.13.

Show that for nonzero a and b, the function $\sqrt{x_1^2\slash a^2+x_2^2\slash b^2}$ defines a norm on the 2-dimensional real vector space.

4.14.

Show that the norms on $\mathbf R^2$ defined by

$$ \Vert x\Vert =\sqrt{x_1^2+x_2^2}\qquad \text {and}\qquad \Vert x\Vert '=\sqrt{x_1^2\slash a^2+x_2^2\slash b^2}, $$

for $a>b>0$, satisfy the inequalities

$$ \genfrac{}{}{0.4pt}{}{1}{a}\Vert x\Vert \le \Vert x\Vert '\le \genfrac{}{}{0.4pt}{}{1}{b}\Vert x\Vert ,\qquad \text {for all } x\in \mathbf R^2. $$

4.15.

Prove that the set

$$ S=\{(c_1,\ldots , c_n):\sum _k|c_k|=1\} $$

is closed and bounded in $\ell _2^n$.

4.16.

Show that if T is an isomorphism of normed spaces X and Y, then both T and $T^{-1}$ are bounded operators.

4.17.

Show that every space $\ell _p^n$ for $0<p<1$ is isomorphic to $\ell _\infty ^n$. Conclude that all spaces $\ell _p^n$, $p\in (0,\infty ]$, are isomorphic.

4.18.

Show that two equivalent norms (cf. Definition 4.4) on a vector space define equivalent operator norms.

4.19.

Let $X_0$ be a proper subspace of the vector space X over the field $\mathbf F$ and $x\in X\setminus X_0$. Show that every vector $z\in X\setminus X_0$ can be written in the form

$$ z=\alpha (x-y),\qquad \text {where }\alpha \ne 0 \text { and }y\in X_0. $$

4.20.

Show that $\Vert f\Vert =1$ for the functional f from Example 4.9.

4.21.

Let x be a continuous nonnegative function on [0, 1] such that $x(0)=0$ and $x(t)\le 1$ for all $t\in [0,1]$. Show that $\int _0^1x(t)\, dt<1$.