Weak Convergence Methods for Nonlinear Partial Differential ...

Corollary 2.5 Let X be a normed space, x n ⇀ x. Then there exists a sequence
of convex combinations
y n =
such that ‖y n − x‖ → 0.
N(n)
∑
i=n
N(n)
∑
λ (n)
i x i , λ (n)
i ≥ 0,
i=n
λ (n)
i = 1
Proof. Let V n = co(x n , x n+1 , ...). As x ∈ V n there exists λ n k , ..., λn N(n) , ∑ λ n i = 1
such that
∥ ∥∥∥∥∥ N(n)
∑
λ n i − x
∥ ≤ 1 n .
i=n
□
Corollary 2.6 The norm ‖ · ‖ X on X (resp. ‖ · ‖ X ′
weakly*) lower semicontinuous.
on X ′ ) is weakly (resp.
Proof. Suppose x n ⇀ x in X. Let R = lim inf ‖x n ‖. Then for ε > 0 ∃ subsequence
such that ‖x nk ‖ ≤ R + ε and thus x ∈ B R+ε . As ε was arbitrary,
‖x‖ ≤ R = lim inf
n→∞
‖x n‖.
If ϕ ∗ n ⇀ ϕ in X ′ , then lim inf ‖ϕ n ‖ ≥ lim inf ϕ n (x) = ϕ(x) for any x ∈ X with
‖x‖ = 1. Passing to the supremum over those x yields lim inf ‖ϕ n ‖ ≥ ‖ϕ‖. □
Weakly(*) convergent sequences are necessarily bounded:
Theorem 2.7 Suppose x n
(∗)
⇀ x. Then (x n ) is bounded in X (resp. X ′ ).
In order to check if a sequence (x n ) converges weakly/weakly*, it is often convenient
to consider particularly simple objects (test functions) in X ′ resp. X on
which to test (x n ). This is often made possible by the following result.
Theorem 2.8 A sequence (x n ) converges weakly to x if and only if
• (x n ) is bounded and
• ϕ(x n ) → ϕ(x) ∀ϕ ∈ A where span A = X ′ .
An analogous statement holds for weak*-convergence.
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