Exercise 26, § 3

Let X be a compact topological space and E a complex Banach space, whose norm is denoted $\|\cdot \|_E$. We denote $\mathscr{C}(X,E)$ the complex vector space of continuous functions from X into E.

a) Endowed with the norm

$\|f\|=$ sup$_{x\in X}\|f(x)\|_E$,

the space $\mathscr{C}(X,E)$ is a Banach space.

b) Let B be the vector subspace of $\mathscr{C}(X,E)$ generated by the functions $x\mapsto f(x)y$, where $f\in \mathscr{C}(X)$ and $y\in E$. Now prove that B is dense in $\mathscr{C}(X,E)$.

In what follows, we consider a commutative Banach algebra A.

c) Endowed with the multiplication $(f g)(x) =f(x)g(x)$ for every $x\in X$, the Banach space $\mathscr{C}(X,A)$ is a commutative Banach algebra.

d) Every character $\chi \in \mathsf{X}(\mathscr{C}(X,A))$ is of the form $f\mapsto \widetilde{\chi}(f(x))$, where $\widetilde{\chi}\in \mathsf{X}(A)$ and $x\in X$. (First treat the case where A is unital and $\|1\|= 1$ ; consider the restriction of $\chi$ to $\mathscr{C}(X)\cdot 1$ and to the image of the homomorphism $A\rightarrow \mathscr{C}(X,A)$ which sends $a$ to the constant function $x\mapsto a$).

e) The mapping from $\mathsf{X}(A)\times X$ into $\mathsf{X}(\mathscr{C}(X,A))$ which to $(\widetilde{\chi , x})$ associates the character $f\mapsto \widetilde{\chi}(f(x))$ is a homeomorphism.