On the global shape of continuous convex functions on Banach spaces

Abstract

We make some remarks on the global shape of continuous convex functions defined on a Banach space Z. Among other results we prove that if Z is separable then for every continuous convex function f : Z → R there exist a unique closed linear subspace Y f of Z such that, for the quotient space X f := Z/Y f and the natural projection π : Z → X f , the function f can be written in the formwhere ℓ f ∈ X * and ϕ : X f → R is a convex function such that limt→∞ ϕ(x + tv) = ∞ for every x, v ∈ X f with v = 0. This kind of result is generally false if Z is nonseparable (even in the Hilbertian case Z = ℓ2(Γ) with Γ an uncountable set).Definition 1. Let Z be a Hilbert space. We will say that a function f : Z → R is essentially coercive provided that there exists a linear function ℓ : Z → R such that lim |z|→∞ (f (z) − ℓ(z)) = ∞.

On the global shape of convex functions on locally convex spaces

“…Let us see the relationships among the sets and functions considered in [1, and those introduced and used previously. As in [1], in the sequel X (:= Z) is a Banach space.…”

“…5]; more precisely c f attains its infimum on Z/Y f instead of taking values in R + . In fact, in general it is not possible to obtain c f : Z/Y f → R + with the desired properties in [1,Th. 5].…”

“…With respect to [1,Th. 4], first observe that the space X f [:= span(Im ∂f − Im ∂f )] is not closed in general; for example, for the function ϕ : ℓ 2 → R defined by ϕ(x) := ∞ n=1 |x n | 2 /2 n (on page 2 of [1]) one has Im ∂ϕ = {(y n ) n≥1 ∈ ℓ 2 | (2 n y n ) n≥1 ∈ ℓ 2 } and dom ϕ * = {(y n ) n≥1 ∈ ℓ 2 | (2 n/2 y n ) n≥1 ∈ ℓ 2 }.…”

“…Having in view the statements of Theorems 5 and 6 in [1], it is worth observing that for x 0 ∈ dom f , u ∈ X and u * ∈ X * one has…”

“…As in [1,Def. 3], we say that f is directionally coercive if lim t→∞ f (x + tu) = ∞ for all x ∈ X and u ∈ X \ {0}, and f is essentially directionally coercive if f − x * is directionally coercive for some x * ∈ X * .…”