Smooth Norms and Approximation in Banach Spaces of the Type 𝒞(k)

Polyhedral norms on non-separable Banach spaces

“…For example, if K is compact and the derived set K (ω 1 ) of order ω 1 is empty, then K (β) is empty for some β < ω 1 and so K = α<β (K (α) \ K (α+1) ) is σ -discrete. The idea behind the proof of the following theorem is based on a result in [13]. Theorem 11.…”

Using boundaries to find smooth norms

“…The second lemma gives a sufficient condition for when || · || φ on X is C k smooth. It uses the notion of local dependence on finitely many coordinates and generalises [9,Lemma 5.3]. Lemma 1.7.…”

“…As Lemma 1.7 appears in [9,Lemma 5.3], X is taken to be a closed subspace of ℓ ∞ (B) and Π is the identity. The proof uses the fact that each coordinate map x → |x(t)| is C ∞ smooth on the set where it is non-zero and uses the implicit function theorem to show that || · || φ is also C ∞ smooth.…”

Locally uniformly convex norms in Banach spaces and their duals

“…In the special case of spaces Z = C(K), this implication has been established in [8]. It is also unknown whether Fréchet-renormability of Z implies LUR renormability.…”