Some Applications

Define measures $\mu_{1}, \mu_{2}$ on the unit circle by

and find the range of the measure $\mu = \left(\mu_{1}, \mu_{2}\right)$.

Construct two functions $f$ and $g$ on $[0,1]$ with the following property: If

then the range of $\mu$ is the square with vertices at $(1,0),(0,1),(-1,0),(0,-1)$.

Suppose that the hypotheses of Theorem 5.9 are satisfied, that $\phi \in C(S), \phi > 0$, $g \in C(K)$, and $\left|g\right| < \left.\phi \right|_{K}$. Prove that there exists $f \in Y$ such that $\left.f\right|_{K} = g$ and $\left|f\right| < \phi$ on S. Hint: Apply Theorem 5.9 to the space of all functions $f / \phi$, with $f \in Y$.

Supply the details of the proof that every extreme point of $P$ has its support at a single point. (This refers to the end of proof of Theorem 5.10).

Prove the analogues of Theorems 1.10 to 1.12 that are alluded to in Section 5.12. (Do not assume that $G$ is commutative).

Suppose $G$ is a topological group and $H$ is the largest connected subset of $G$ that contains the identity element $e$. Prove that $H$ is a normal subgroup of $G$, that is, a subgroup that satisfies $x^{-1} H x = H$ for every $x \in G$. Hint: If $A$ and $B$ are connected subsets of $G$, so are $A B$ and $A^{-1}$.

Prove that every open subgroup of a topological group is closed. (The converse is obviously false).

Suppose $m$ is the Haar measure of a compact group $G$, and $V$ is a nonempty open set in $G$. Prove that $m(V) > 0$.

Put $e_{n}(\theta ) = e^{i n \theta }$. Let $L^{2}$ refer to the Haar measure of the unit circle. Let $A$ be the smallest closed subspace of $L^{2}$ that contains $e_{n}$ for $n = 0,1,2, \ldots$, let $B$ be the smallest closed subspace of $L^{2}$ that contains $e_{-n}+n e_{n}$ for $n = 1,2,3, \ldots$ Prove the following:

Suppose $X$ is a Banach space, $P \in \mathscr {B}(X), Q \in \mathscr {B}(X)$, and $P$ and $Q$ are projections.

Suppose $P$ and $Q$ are projections in a vector space $X$.

Prove that the translation operators $\tau_{s}$ used in Example 5.19 satisfy the continuity property described in Section 5.17. Explicitly, prove that

if $r \rightarrow s$ and $g \rightarrow f$ in $L^{1}$.

Use the following example to show that the compactness of $G$ cannot be omitted from the hypotheses of Theorem 5.18. Take $X = L^{1}$ on the real line $\mathbb {R}$, relative to Lebesgue measure; $f \in Y$ if and only if $\int_{\mathbb {R}} f = 0 ; G = \mathbb {R}$ with the usual topology; $G$ acts on $L^{1}$ by translation: $\left(\tau_{s} f\right)(x) = f(s+x)$. The joint continuity property is satisfied (see Exercise 12), $\tau_{s} Y = Y$ for every $s$, and $Y$ is complemented in $X$. Yet there is no projection of $X$ onto $Y$ (continuous or not) that commutes with every $\tau_{s}$.

Suppose $S$ and $T$ are continuous linear operators in a topological vector space, and

Prove that $T$ has closed range. (See Theorem 5.16 for the case $S = I$).

Suppose $A$ is a closed subspace of $C(S)$, where $S$ is a compact Hausdorff space; suppose $\mu$ is an extreme point of the unit ball of $A^{\perp }$; and suppose $f \in C(S)$ is a real function such that

for every $g \in A$. Prove that $f$ is then constant on the support of $\mu$. (Compare with Theorem 5.7). Show, by an example, that the conclusion is false if the word "real" is omitted from the hypotheses.

Suppose $X$ is a vector space, $E \subset X, T: \operatorname {co}(E) \rightarrow X$ is affine, and $T(E) \subset E$. Prove that $T(\operatorname {co}(E)) \subset \operatorname {co}(E)$. (This was tacitly used in the proof of Theorem 5.11).

If $X$ and $Y$ are vector spaces and $T: X \rightarrow Y$ is affine, prove that $T-T(0)$ is linear.

Suppose $K$ is a compact set in a Fréchet space $X$ and $f: X \rightarrow K$ is continuous. Prove that $f$ fixes some point of $K$.

Do the same if $\Omega$ is a convex open set in $X, \Omega \supset K$, and $f: \Omega \rightarrow K$ is continuous.

Prove the existence of a continuous function $f$ on $I = [0,1]$ which satisfies the equation

for all $x \in I$. Hint: Denoting the right side by $(T f)(x)$, show that the set $\left\{ T f: f \in C(I)\right\}$ is uniformly bounded and equicontinuous and that its closure is therefore compact in $C(I)$. Apply Schauder's fixed point theorem (via Exercise 18).