Examples of Banach Space Techniques

Let $X$ consist of two points $a$ and $b$, put $\mu (\left\{ a\right\} ) = \mu (\left\{ b\right\} ) = \frac{1}{2}$, and let $L^{p}(\mu )$ be the resulting real $L^{p}$-space. Identify each real function $f$ on $X$ with the point $(f(a), f(b))$ in the plane, and sketch the unit balls of $L^{p}(\mu )$, for $0 < p \leq \infty$. Note that they are convex if and only if $1 \leq p \leq \infty$. For which $p$ is this unit ball a square? A circle? If $\mu (\left\{ a\right\} ) \neq \mu (b)$, how does the situation differ from the preceding one?

Prove that the unit ball (open or closed) is convex in every normed linear space.

If $1 < p < \infty$, prove that the unit ball of $L^{p}(\mu )$ is strictly convex; this means that if

then $\left\| h\right\|_{p} < 1$. (Geometrically, the surface of the ball contains no straight lines). Show that this fails in every $L^{1}(\mu )$, in every $L^{\infty }(\mu )$, and in every $C(X)$. (Ignore trivialities, such as spaces consisting of only one point).

Let $C$ be the space of all continuous functions on [0,1], with the supremum norm. Let $M$ consist of all $f \in C$ for which

Prove that $M$ is a closed convex subset of $C$ which contains no element of minimal norm.

Let $M$ be the set of all $f \in L^{1}([0,1])$, relative to Lebesgue measure, such that

Show that $M$ is a closed convex subset of $L^{1}([0,1])$ which contains infinitely many elements of minimal norm. (Compare this and Exercise 4 with Theorem 4.10).

Let $f$ be a bounded linear functional on a subspace $M$ of a Hilbert space $H$. Prove that $f$ has a unique norm-preserving extension to a bounded linear functional on $H$, and that this extension vanishes on $M^{\perp }$.

Construct a bounded linear functional on some subspace of some $L^{1}(\mu )$ which has two (hence infinitely many) distinct norm-preserving linear extensions to $L^{1}(\mu )$.

Let $X$ be a normed linear space, and let $X^{*}$ be its dual space, as defined in Sec. 5.21, with the norm

Let $c_{0}, \ell^{1}$, and $\ell^{\infty }$ be the Banach spaces consisting of all complex sequences $x = \left\{ \xi_{i}\right\}$, $i = 1,2,3, \ldots$, defined as follows:

$c_{0}$ is the subspace of $\ell^{\infty }$ consisting of all $x \in \ell^{\infty }$ for which $\xi_{i} \rightarrow 0$ as $i \rightarrow \infty$. Prove the following four statements.

If $\sum \alpha_{i} \xi_{i}$ converges for every sequence $\left\{ \xi_{i}\right\}$ such that $\xi_{i} \rightarrow 0$ as $i \rightarrow \infty$, prove that $\sum \left|\alpha_{i}\right| < \infty$.

For $0 < \alpha \leq 1$, let Lip $\alpha$ denote the space of all complex functions $f$ on $[a, b]$ for which

Prove that Lip $\alpha$ is a Banach space, if $\left\| f\right\| = \left|f(a)\right|+M_{f}$; also, if

(The members of Lip $\alpha$ are said to satisfy a Lipschitz condition of order $\alpha$).

Let $K$ be a triangle (two-dimensional figure) in the plane, let $H$ be the set consisting of the vertices of $K$, and let $A$ be the set of all real functions $f$ on $K$, of the form

Show that to each $\left(x_{0}, y_{0}\right) \in K$ there corresponds a unique measure $\mu$ on $H$ such that

(Compare Sec. 5.22). Replace $K$ by a square, let $H$ again be the set of its vertices, and let $A$ be as above. Show that to each point of $K$ there still corresponds a measure on $H$, with the above property, but that uniqueness is now lost.

Can you extrapolate to a more general theorem? (Think of other figures, higher dimensional spaces).

Let $\left\{ f_{n}\right\}$ be a sequence of continuous complex functions on a (nonempty) complete metric space $X$, such that $f(x) = \lim f_{n}(x)$ exists (as a complex number) for every $x \in X$.

Let $C$ be the space of all real continuous functions on $I = [0,1]$ with the supremum norm. Let $X_{n}$ be the subset of $C$ consisting of those $f$ for which there exists a $t \in I$ such that $\left|f(s)-f(t)\right| \leq n\left|s-t\right|$ for all $s \in I$. Fix $n$ and prove that each open set in $C$ contains an open set which does not intersect $X_{n}$. (Each $f \in C$ can be uniformly approximated by a zigzag function $g$ with very large slopes, and if $\left\| g-h\right\|$ is small, $h \notin X_{n}$). Show that this implies the existence of a dense $G_{\delta }$ in $C$ which consists entirely of nowhere differentiable functions.

Let $A = \left(a_{i j}\right)$ be an infinite matrix with complex entries, where $i, j = 0,1,2, \ldots A$ associates with each sequence $\left\{ s_{j}\right\}$ a sequence $\left\{ \sigma_{i}\right\}$, defined by

provided that these series converge. Prove that $A$ transforms every convergent sequence $\left\{ s_{j}\right\}$ to a sequence $\left\{ \sigma_{i}\right\}$ which converges to the same limit if and only if the following conditions are satisfied:

Suppose $X$ and $Y$ are Banach spaces, and suppose $\Lambda$ is a linear mapping of $X$ into $Y$, with the following property: For every sequence $\left\{ x_{n}\right\}$ in $X$ for which $x = \lim x_{n}$ and $y = \lim \Lambda x_{n}$ exist, it is true that $y = \Lambda x$. Prove that $\Lambda$ is continuous.

This is the so-called "closed graph theorem". Hint: Let $X \oplus Y$ be the set of all ordered pairs $(x, y), x \in X$ and $y \in Y$, with addition and scalar multiplication defined componentwise. Prove that $X \oplus Y$ is a Banach space, if $\left\| (x, y)\right\| = \left\| x\right\| +\left\| y\right\|$. The graph $G$ of $\Lambda$ is the subset of $X \oplus Y$ formed by the pairs $(x, \Lambda x), x \in X$. Note that our hypothesis says that $G$ is closed; hence $G$ is a Banach space. Note that $(x, \Lambda x) \rightarrow x$ is continuous, one-to-one, and linear and maps $G$ onto $X$.

Observe that there exist nonlinear mappings (of $\mathbb {R}^{1}$ onto $\mathbb {R}^{1}$, for instance) whose graph is closed although they are not continuous: $f(x) = 1 / x$ if $x \neq 0, f(0) = 0$.

If $\mu$ is a positive measure, each $f \in L^{\infty }(\mu )$ defines a multiplication operator $M_{f}$ on $L^{2}(\mu )$ into $L^{2}(\mu )$, such that $M_{f}(g) = f g$. Prove that $\left\| M_{f}\right\| \leq \left\| f\right\|_{\infty }$. For which measures $\mu$ is it true that $\left\| M_{f}\right\| = \left\| f\right\|_{\infty }$ for all $f \in L^{\infty }(\mu )$? For which $f \in L^{\infty }(\mu )$ does $M_{f}$ map $L^{2}(\mu )$ onto $L^{2}(\mu )$?

Suppose $\left\{ \Lambda_{n}\right\}$ is a sequence of bounded linear transformations from a normed linear space $X$ to a Banach space $Y$, suppose $\left\| \Lambda_{n}\right\| \leq M < \infty$ for all $n$, and suppose there is a dense set $E \subset X$ such that $\left\{ \Lambda_{n} x\right\}$ converges for each $x \in E$. Prove that $\left\{ \Lambda_{n} x\right\}$ converges for each $x \in X$.

If $s_{n}$ is the $n$th partial sum of the Fourier series of a function $f \in C(T)$, prove that $s_{n} / \log n \rightarrow 0$ uniformly, as $n \rightarrow \infty$, for each $f \in C(T)$. That is, prove that

On the other hand, if $\lambda_{n} / \log n \rightarrow 0$, prove that there exists an $f \in C(T)$ such that the sequence $\left\{ s_{n}(f ; 0) / \lambda_{n}\right\}$ is unbounded. Hint: Apply the reasoning of Exercise 18 and that of Sec. 5.11, with a better estimate of $\left\| D_{n}\right\|_{1}$ than was used there.

Suppose $E \subset \mathbb {R}^{1}$ is measurable, and $m(E) = 0$. Must there be a translate $E+x$ of $E$ that does not intersect $E$? Must there be a homeomorphism $h$ of $\mathbb {R}^{1}$ onto $\mathbb {R}^{1}$ so that $h(E)$ does not intersect $E$?

Suppose $f \in C(T)$ and $f \in \operatorname {Lip} \alpha$ for some $\alpha > 0$. (See Exercise 11). Prove that the Fourier series of $f$ converges to $f(x)$, by completing the following outline: It is enough to consider the case $x = 0$, $f(0) = 0$. The difference between the partial sums $s_{n}(f ; 0)$ and the integrals

tends to 0 as $n \rightarrow \infty$. The function $f(t) / t$ is in $L^{1}(T)$. Apply the Riemann-Lebesgue lemma. More careful reasoning shows that the convergence is actually uniform on $T$.