theorem 16.70 The direct method

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:2172 · p. 655

Rests on

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 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - 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 definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on equation 3.51 eq:set-indexed ¶ ↺
proves proof ch:14-calculus-of-variations@proof-34 ¶

Supports

depends_on lemma 16.72 Convexity implies weak lower semicontinuity ¶
- depends_on theorem A.448 Tonelli ¶
- depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
- depends_on theorem 16.73 Tonelli ¶
depends_on theorem A.477 Douglas; Radó ¶
depends_on theorem A.448 Tonelli ¶ ↺
depends_on theorem 16.81 Douglas; Radó ¶
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`	→	Hilbert space	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2184
`depends_on`	→	Compact set	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2184
`depends_on`	←	Convexity implies weak lower semicontinuity	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2234
`depends_on`	←	Douglas; Radó	declared	appendices/A-long-proofs.tex:23495
`depends_on`	←	Tonelli	declared	appendices/A-long-proofs.tex:22139
`depends_on`	←	Douglas; Radó	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2573
`depends_on`	←	Tonelli	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2271
`proves`	←	ch:14-calculus-of-variations@proof-34	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2187