数学代写|Math5052 Functional Analysis

Let $X$ be a normed linear space. For any convex $B \subset X$, say that a subset $F \subset B$ is a face of $B$ if, given $x, y \in B$ and $0<\theta<1$ with $\theta x+(1-\theta) y \in F$, one may conclude that $x, y \in F$.
a. Suppose $f$ is a bounded linear functional on $X$ and $B \subset X$ is a convex subset such that $\beta \stackrel{\text { def }}{=} \sup {f(x): x \in B}$ is finite. Define
$$
F \stackrel{\text { def }}{=}{x \in B: f(x)=\beta}
$$
Prove that $F$ is a face of $B$.
b. Suppose $B$ is a convex set, $F \subset B$ is a face of $B$, and $G \subset F$ is any subset. Prove that $G$ is a face of $F$ if and only if $G$ is a face of $B$.

Let $X$ be a linear space. For any convex $B \subset X$, say that $e \in B$ is an extreme point of $B$ iff
$$
(\forall x, y \in B)(\forall \theta \in(0,1)) \quad e=\theta x+(1-\theta) y \Rightarrow e=x=y .
$$
a. Suppose $B$ is an open convex set in a normed linear space $X$. Prove that $B$ has no extreme points.
b. Suppose $B$ is a compact convex set in a Banach space $X$. Prove that if $B$ is non-empty then $B$ contains an extreme point.
(Hint: Apply Zorn’s lemma to the collection of closed non-empty faces of $B$ partially ordered by $F_1 \leq F_2$ iff $F_2$ is a face of $F_1$. Show that any maximal element contains a single point of $B$, which is therefore an extreme point.)

Let $X$ be a linear space and $A \subset X$ any subset. Define the convex hull of $A$ to be
$$
\operatorname{ch}(A) \stackrel{\text { def }}{=}{\theta x+(1-\theta) y: x, y \in A ; 0 \leq \theta \leq 1} .
$$
If $X$ is a normed linear space, define the closed convex hull of $A$ to be the closure of ch $(A)$, and denote it by $\overline{\operatorname{ch}}(A)$.
a. Prove that if $A \subset B \subset X$, then $\overline{\operatorname{ch}}(A) \subset \overline{\operatorname{ch}}(B)$.
b. Prove that if $A$ is a closed convex set, then $A=\overline{\operatorname{ch}}(A)$.

Let $X$ be a Banach space and suppose that $A \subset X$ is compact and convex. Let $E \subset A$ be the set of extreme points of $A$ as defined in exercise 25. Prove that $A=\overline{\operatorname{ch}}(E)$.
Hint: use exercises 23 and 26.
Note: this is the Krein-Milman theorem.

Let $\langle f, g\rangle \stackrel{\text { def }}{=} \int_0^1 f(x) \overline{g(x)} d x$ be the usual inner product in $L^2([0,1])$. Prove that $C([0,1])$ is not a Hilbert space with respect to this inner product and its derived norm.

Suppose that $\left{x_n\right}$ is a sequence in a Hilbert space $H$. Suppose $\left|x_n\right| \rightarrow|x|$ and $(\forall y \in H)\left\langle x_n, y\right\rangle \rightarrow$ $\langle x, y\rangle$ as $n \rightarrow \infty$. Prove that $\left|x_n-x\right| \rightarrow 0$ as $n \rightarrow \infty$.