lemma 16.72 Convexity implies weak lower semicontinuity

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:2224 · p. 656

Rests on

depends_on equation 16.11 eq:calcvar-basic-functional ¶
depends_on theorem 16.70 The direct method ¶
- depends_on definition 12.2 Hilbert space ¶
  - depends_on definition 5.18 Inner product ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on equation 5.44 eq:lin-norm-assoc ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- proves proof ch:14-calculus-of-variations@proof-34 ¶
proves proof ch:14-calculus-of-variations@proof-35 ¶

Supports

depends_on theorem A.448 Tonelli ¶
depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
depends_on theorem 16.73 Tonelli ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:calcvar-basic-functional	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2234
`depends_on`	→	The direct method	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2234
`depends_on`	←	Tonelli	declared	appendices/A-long-proofs.tex:22139
`depends_on`	←	Convexity in $q$ implies weak lower semicontinuity	declared	appendices/A-long-proofs.tex:22444
`depends_on`	←	Tonelli	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2271
`proves`	←	ch:14-calculus-of-variations@proof-35	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2237