488 CHAPTER 18. TOPOLOGICAL VECTOR SPACES

However, for all n large enough, the first condition is satisfied. Consequently, for all n largeenough, φ (xn)≥ l +δ ≥ l. Thus

lim infn→∞

φ (xn)≥ l

and since l < φ (x) is arbitrary, it follows

lim infn→∞

φ (xn)≥ φ (x) .

Next suppose the condition about the liminf. If epi(φ) is not closed, then there exists(x, l) /∈ epi(φ) which is a limit point of points of epi(φ) , Thus there exists (xn, ln)∈ epi(φ)such that (xn, ln)→ (x, l) and so

l = lim infn→∞

ln ≥ lim infn→∞

φ (xn)≥ φ (x) ,

contradicting (x, l) /∈ epi(φ). This proves the lemma.

Definition 18.2.10 Let φ be convex and defined on X , a Banach space. Then φ is said tobe weakly lower semicontinuous if epi(φ) is closed in X×R where a basis for the topologyof X×R consists of sets of the form U× (a,b) for U a weakly open set in X.

Theorem 18.2.11 Let φ be a lower semicontinuous convex functional as described in Def-inition 18.2.7 and let X be a real Banach space. Then φ is also weakly lower semicontinu-ous.

Proof: By Lemma 18.2.9 epi(φ) is closed in X×R with the strong topology as well asbeing convex. Letting (z, l) /∈ epi(φ) , it follows from Theorem 18.2.5 and Lemma 18.2.8there exists (x∗,α) ∈ X ′×R such that for some c

x∗ (z)+αl > c > x∗ (x)+αβ

whenever β ≥ φ (x) . Consider B{(x∗,α)} ((z, l) ,r) where r is chosen so small that if (y,γ) ∈B{(x∗,α)} ((z, l) ,r) , then

x∗ (y)+αγ > c.

This shows that the complement of epi(φ) is weakly open and this proves the theorem.

Corollary 18.2.12 Let φ be a lower semicontinuous convex functional as described in Def-inition 18.2.7 and let X be a real Banach space. Then if xn converges weakly to x, it followsthat

φ (x)≤ lim infn→∞

φ (xn) .

Proof: Let l < φ (x) so that (x, l) /∈ epi(φ). Then by Theorem 18.2.11 there existsB× (−∞, l +δ ) such that B is a weakly open set in X containing x and

B× (−∞, l +δ )⊆ epi(φ)C .