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The direct method in a reflexive Banach space

Statement

Assume the ultrafilter lemma, DC and HB. Let $X$ be a real reflexive Banach space (Reflexivity is surjectivity of the canonical map), let $A \subseteq X$ be nonempty and weakly sequentially closed (Weak convergence of nets and sequences), and let $I : A \to (- \infty, + \infty]$ be proper, coercive on $A$ and weakly sequentially lower semicontinuous on $A$ (Proper, coercive and weakly lower semicontinuous extended-real functionals). Then $I$ attains its infimum on $A$ : there exists $u_{0} \in A$ with $I (u_{0}) = \inf_{A} I$ . The admissible set may be taken convex and norm closed in $X$ , by Norm closed convex iff weakly closed.

Facts & Assumptions

Given: The ultrafilter lemma (The ultrafilter extension principle (UL/BPI)), the principle of dependent choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) and HB (The real dominated-extension principle as an additional hypothesis over ZF); a real reflexive Banach space $X$ (Reflexivity is surjectivity of the canonical map); a nonempty set $A \subseteq X$ that is weakly sequentially closed; and a proper extended-real functional $I : A \to (- \infty, + \infty]$ , coercive on $A$ and weakly sequentially lower semicontinuous on $A$ (Proper, coercive and weakly lower semicontinuous extended-real functionals). Write $α : = \inf_{A} I$ in the complete extended order specified by Proper, coercive and weakly lower semicontinuous extended-real functionals.

[F1]

Properness gives $α < + \infty$ ; moreover for every real $ε > 0$ there is $u \in A$ with $I (u) < α + ε$ when $α \in R$ , and when $α = - \infty$ there is for every $M \in R$ a point $u \in A$ with $I (u) \leq M$ (Proper, coercive and weakly lower semicontinuous extended-real functionals, Greatest lower bound (infimum)).

[F2]

Coercivity bounds finite-level sequences: every sequence $(u_{j}) \subseteq A$ with $\sup_{j} I (u_{j}) < + \infty$ is norm bounded, and in particular every minimising sequence with $I (u_{j}) \to \inf_{A} I < + \infty$ is norm bounded (Coercivity bounds every finite-level sequence).

[F3]

Under the ultrafilter lemma, DC and HB, every norm-bounded sequence in a real reflexive Banach space has a subsequence converging weakly to a point of $X$ (A bounded sequence in a reflexive Banach space has a weakly convergent subsequence).

[F4]

If $A$ is weakly sequentially closed, $(u_{j}) \subseteq A$ and $u_{j} ⇀ u$ , then $u \in A$ (Weak closedness keeps the direct-method limit admissible).

[F5]

If $(u_{j}) \subseteq A$ is minimising, $u_{j} ⇀ u \in A$ and $I$ is weakly sequentially lower semicontinuous at $u$ , then $I (u) = \inf_{A} I$ (The liminf passage makes the weak limit a minimiser).

[F6]

Dependent choice implies countable choice, so a countable sequence of independent nonempty selections can be made along $N$ (Dependent choice implies countable choice).

[F7]

Under HB, which is assumed here, a convex subset of a real or complex normed space is norm closed if and only if it is weakly closed (Norm closed convex iff weakly closed). A weakly closed set is weakly sequentially closed, so an admissible set that is convex and norm closed satisfies the theorem hypothesis under the stated choice principles.

[F8]

Under HB and Countable Choice (supplied here by DC), in the convex case the weak-lower-semicontinuity hypothesis of the theorem is verified by convexity plus norm lower semicontinuity: a convex norm-lower-semicontinuous functional on a convex set is weakly sequentially lower semicontinuous (A convex norm-lower-semicontinuous functional is weakly lower semicontinuous). This is the role of that lemma for the present theorem and for its convex applications.

Proof

technique · direct, by selecting a minimising sequence, extracting a weakly convergent subsequence and passing to the limit

1.1F1F6given

A minimising sequence. If $α \in R$ , [F1] supplies for each $j \in N$ a point $u_{j} \in A$ with $I (u_{j}) < α + 2^{- j}$ ; if $α = - \infty$ , [F1] supplies $u_{j} \in A$ with $I (u_{j}) \leq - j$ . In both cases $(u_{j}) \subseteq A$ satisfies $I (u_{j}) \to α$ , so it is a minimising sequence. The countably many selections are licensed by [F6].

2.1F2step 1.1

Boundedness. In the finite case $I (u_{j}) < α + 1 < + \infty$ for all $j$ ; in the case $α = - \infty$ one has $I (u_{j}) \leq 0$ for all $j$ . Hence $\sup_{j} I (u_{j}) < + \infty$ , and [F2] makes $(u_{j})$ norm bounded.

3.1F3F4step 2.1

A weakly convergent subsequence with admissible limit. By [F3] there are a strictly increasing sequence $j_{k}$ and a point $u_{0} \in X$ with $u_{j_{k}} ⇀ u_{0}$ ; the subsequence lies in $A$ , which is weakly sequentially closed, so [F4] gives $u_{0} \in A$ .

4.1F5F7F8step 3.1∎

The limit is a minimiser, and the convex special case. The subsequence $(u_{j_{k}})$ is still minimising, $I (u_{j_{k}}) \to α$ , and $u_{j_{k}} ⇀ u_{0} \in A$ , so the weak lower semicontinuity hypothesis and [F5] give $I (u_{0}) = \inf_{A} I$ : the infimum is attained on $A$ . If in addition $A$ is convex and closed in the norm topology, the HB-form of [F7] and the weak-closed-to-weakly-sequentially-closed passage make $A$ weakly sequentially closed, so the theorem applies to that admissible set; and in the convex case the weak lower semicontinuity hypothesis itself is supplied by [F8] whenever $I$ is convex and norm lower semicontinuous. No stronger choice principle than the HB assumed here is needed for these clauses.

Depends on

Used by

A coercive functional need not attain without weak lower semicontinuity Counterexample
A minimising sequence need not converge strongly Counterexample
A norm-closed nonconvex set need not be weakly closed Counterexample
The direct method for convex integral functionals Theorem
The direct method on a weakly closed constraint set Theorem

Dependency tree · two levels

44 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2026 author manuscript; complete 392-page archived text) (standard reference, not scraped)
Francesco Paolo Maiale (course by Giovanni Alberti), Lecture Notes Calculus of Variations A, University of Pisa (last update 21 August 2019; complete 149-page notes) (standard reference, not scraped)
Viktor Grigoryan, Math 246B Partial Differential Equations, UCSB 2011 (complete 31-page course notes) (standard reference, not scraped)