Spaces of Continuous Functions

Prove that $C(X)$ is a Banach space for any compact space $X$.

Prove that $C(X)$ is a separable space if $X$ is a compact subset of $\mathbb {R}^{n}$.

Call a linear functional $F$ on $C(X)$ positive if $F(f)\geq 0$ for all nonnegative functions $f\in C(X)$. Prove that every positive linear functional $F$ is continuous, and its norm is equal to $F(1)$, where $1$ is the function identically equal to $1$ on $X$.

Prove that any functional $F\in C'(X)$ can be written in the form $F=F_1-F_2$, where $F_1$ and $F_2$ are positive functionals (see Problem 394).

Let $X$ be a compact metric space, and $F\in C'(X)$ a positive functional (see Problem 394). Define $\mu (K)=\inf_{\; \mathbb {1}_{K}\leq \varphi \leq 1}F(\varphi )$ for a compact set $K\subset X$, and $\mu (E)=\sup_{K\subset E}\mu (K)$ ($K$ compact) for a Borel set $E\subset X$. Prove that $\mu$ is a countably additive measure.

Prove that a sequence of functionals $F_n(f)=\int_0^1 f(x)\, dg_n(x)$, with $g_n\in BV[0,1]$, has a weak-* limit functional $F(f)=\int_0^1 f(x)\, dg(x)$ with $g\in BV[0,1]$ if and only if $g_n(x)\to g(x)$ at each point of $[0,1]$ and the variations of the functions $g_n$ are collectively bounded. Here $BV[0,1]$ is the space of functions of bounded variation on $[0,1]$.

Let $\mathcal P$ be the subspace of polynomials in $C[0,1]$. Which of the following linear functionals on $\mathcal P$ admit continuous extension to $C[0,1]$ ($p$ denotes the polynomial $\sum_{k=0}^{\deg p}a_k x^k$):

Let $X$ be a connected compact space. Prove that the unit ball in $C(X)$ has exactly two extreme points.

Prove that the signed point measures $\pm \mu_x$, $x\in X$, defined by the formula $\left\langle \mu_x, f \right\rangle =f(x)$ are the extreme points of the unit ball of $C'(X)$.

Let $X$ be a compact metric space, and $A\subset C(X)$ a closed subalgebra separating points (i.e., for any two distinct points $x_1$ and $x_2$ in $X$ there is a function $\varphi \in A$ such that $\varphi (x_1)\neq \varphi (x_2)$) and containing the function identically equal to $1$.

Let $X$ be an arcwise connected compact metric space. Construct a continuous mapping of $[0,1]$ onto $X$.

Construct a continuous mapping of $[0,1]$ onto the unit square.

Construct an isometric imbedding of $l_p(2,\mathbb {R})$ into $C[0,1]$ with the aid of a continuous mapping of $[0,1]$ onto the unit sphere of $l_p(2,\mathbb {R})$.

Prove that the spaces $C[0,1]\underset {\vee }{\otimes }C[0,1]$ and $C(\square )$ are isomorphic, where $\square$ is the unit square in $\mathbb {R}^{2}$.

Prove that there is an isomorphism $C(X)\underset {\vee }{\otimes }C(Y)\approx C(X\times Y)$ for any compact subsets $X$ and $Y$ of $\mathbb {R}^{n}$.

Let $A:C(X)\to C(Y)$ be an isomorphism of Banach spaces. Prove that $A$ has the form $(Af)(y)=a(y)f(\varphi (y))$, where $a$ is a continuous function on $Y$ taking the values $\pm 1$, and $\varphi$ is a homeomorphism of $Y$ onto $X$.

Prove that the space of all functions of the form $f(x)+g(y)$, where $f,g\in C[0,1]$, is closed in $C(\square )$ ($\square$ is the unit square in $\mathbb {R}^{2}$).

Prove that the space $C[0,1]$ has a countable topological basis $\left\{ f_n(x)\right\}$, i.e., a system of functions $\left\{ f_n(x)\right\}$ such that any $f\in C[0,1]$ can be uniquely represented as a uniformly convergent series $f(x)=\sum_{n=1}^{\infty }c_n f_n(x)$.

Prove that the system of functions $\left\{ e^{2\pi inx}\right\}$, $n\in \mathbb {Z}$, is not a topological basis (see Problem 410) in the space $CP[0,1]$ of all continuous functions on $[0,1]$ with the condition $f(0)=f(1)$.