Approximation by DC Functions and Application to Representation of a Normed Semigroup

Let

be a nonempty compact set of a locally convex space

L L

, and let

C(\Omega) C ( Ω )

be the Banach space of all real-valued continuous functions on

\Omega Ω

endowed with the

\sup sup ⁡

-norm. In this paper, we show first that for every

f\in C(\Omega) f ∈ C ( Ω )

, and for every

\varepsilon>0 ε > 0

, there are continuous affine functions

(g_i)_{i=1}^m, (h_j)_{j=1}^n ( g i ) i = 1 m , ( h j ) j = 1 n

on

L L

for some

m,n\in\mathbb{N} m , n ∈ N

such that

|f(\omega)-[(g_1\vee g_2\vee\cdots\vee{g_m})-(h_1\vee h_2\vee \cdots\vee{h_n})](\omega)|<\varepsilon ∣ f ( ω ) − [ ( g 1 ∨ g 2 ∨ ⋯ ∨ g m ) − ( h 1 ∨ h 2 ∨ ⋯ ∨ h n ) ] ( ω ) ∣ < ε

uniformly for

\omega\in\Omega ω ∈ Ω

. We prove then that if

\Omega=B_{X^*} Ω = B X ∗

, the closed unit ball of

X^* X ∗

of a Banach space

X X

endowed with the

w^* w ∗

-topology, then

C(\Omega)^* C ( Ω ) ∗

is just the dual of the normed semigroup b

(X) ( X )

generated closed balls in

X X