Deville's Master Lemma and Stone's Discreteness in Renorming Theory

Banach spaces

with an equivalent

\sigma(X,F) σ ( X , F )

-lower semicontinuous and locally uniformly rotund norm, for a norming subspace

F\subset X^* F ⊂ X ∗

, are those spaces

X X

that admit countably many families of convex and

\sigma(X,F) σ ( X , F )

-lower semicontinuous functions

\{\varphi_i^n:X \rightarrow {\mathbb R}^+; i \in I_n\}_{n=1}^\infty { φ i n : X → R + ; i ∈ I n } n = 1 ∞

such that there are open subsets

G_i^n \subset \{\varphi_i^n >0\} \cap\{\varphi_j^n =0: j\neq i, j \in I_n\} G i n ⊂ { φ i n > 0 } ∩ { φ j n = 0 : j ≠ i , j ∈ I n }

with

\{G_i^n: i\in I_n, n\in {\mathbb N}\} { G i n : i ∈ I n , n ∈ N }

a basis for the norm topology of

X X

.