Math 2.0

Calculus of variations and optimal control; optimization Definition

Mosco convergence

Builds on Γ-convergenceNormed vector spaceWeak topologyExtended real number lineLimit inferiorLimit superiorContinuous dual space

Definition

Let XX be a real normed space. Write xn⇀xx_n \rightharpoonup x for convergence in the weak topology σ(X,X∗)\sigma(X, X^*) and xn→xx_n \to x for convergence in norm, where X∗X^* is the continuous dual space of XX. Let Fn ⁣:X→[−∞,+∞]F_n \colon X \to [-\infty, +\infty] for n∈ℕn \in \N and F ⁣:X→[−∞,+∞]F \colon X \to [-\infty, +\infty] be functions with values in the extended real line. The sequence (Fn)(F_n) Mosco converges to FF if the following two conditions hold for every x∈Xx \in X.

  1. (Weak lower bound.) For every sequence (xn)(x_n) in XX with xn⇀xx_n \rightharpoonup x, F(x)≤lim inf⁡n→∞Fn(xn).F(x) \le \liminf_{n \to \infty} F_n(x_n).
  2. (Strong recovery sequence.) There is a sequence (xn)(x_n) in XX with xn→xx_n \to x and F(x)≥lim sup⁡n→∞Fn(xn).F(x) \ge \limsup_{n \to \infty} F_n(x_n).

The indicator function of a set K⊆XK \subseteq X in convex analysis is 00 on KK and +∞+\infty outside KK. Subsets Kn⊆XK_n \subseteq X Mosco converge to K⊆XK \subseteq X if their indicator functions do.

Remarks

For sets the two conditions read as follows. If nkn_k is increasing, xk∈Knkx_k \in K_{n_k} and xk⇀xx_k \rightharpoonup x, then x∈Kx \in K. Every x∈Kx \in K is the norm limit of a sequence with xn∈Knx_n \in K_n. The notation F=M-lim⁡nFnF = \operatorname{M\text{-}lim}_n F_n is common. If XX is finite-dimensional, weak and norm convergence agree, and Mosco convergence is Γ-convergence.

Example: let FF be convex and lower semicontinuous, and let Fn=FF_n = F for all nn. Then (Fn)(F_n) Mosco converges to FF. The recovery sequence xn=xx_n = x gives clause 2. For clause 1, a convex lower semicontinuous function is also weakly lower semicontinuous, because its closed sublevel sets are convex and hence weakly closed.

Non-example: in the sequence space ℓ2\ell^2 with unit vectors ene_n, let Kn={en}K_n = \set{e_n} and K={0}K = \set{0}. Clause 1 holds, since en⇀0e_n \rightharpoonup 0. Clause 2 fails at 00: every choice xn∈Knx_n \in K_n is xn=enx_n = e_n, and ∥en−0∥=1\norm{e_n - 0} = 1 for all nn.

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Continuous dual space, concept, no evidence levelContinuous dual spaceconceptExtended real number line, object, no evidence levelExtended real number …objectLimit inferior, concept, no evidence levelLimit inferiorconceptLimit superior, concept, no evidence levelLimit superiorconceptNormed vector space, concept, no evidence levelNormed vector spaceconceptWeak topology, concept, no evidence levelWeak topologyconceptΓ-convergence, concept, no evidence levelΓ-convergenceconceptMosco convergence, definition, no evidence levelMosco convergencedefinition
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  1. Continuous dual space concept
  2. Extended real number line object
  3. Limit inferior concept
  4. Limit superior concept
  5. Normed vector space concept
  6. Weak topology concept
  7. Γ-convergence concept
  8. Mosco convergence definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mosco_convergence, Definition
About this page
TagRNDKD74V17
URIm2://RNDKD74V17
Kinddefinition
WritingW0 stub
MSC49J45, 49J40, 49J53
DefinesMosco convergence
UsesΓ-convergence, Normed vector space, Weak topology, Extended real number line, Limit inferior, Limit superior, Continuous dual space

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