theorem A.448 Tonelli

open in the book · appendices/A-long-proofs.tex:22114 · p. 3013

Rests on

depends_on equation 16.11 eq:calcvar-basic-functional ¶
depends_on lemma 16.72 Convexity implies weak lower semicontinuity ¶
- 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 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on definition 6.24 Metric ¶
    - 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 equation 3.51 eq:set-indexed ¶
  - proves proof ch:14-calculus-of-variations@proof-34 ¶
- proves proof ch:14-calculus-of-variations@proof-35 ¶
depends_on theorem 16.70 The direct method ¶ ↺
proves proof app:A-long-proofs@proof-273 ¶

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type	direction	node	provenance	where
`depends_on`	→	eq:calcvar-basic-functional	declared	appendices/A-long-proofs.tex:22139
`depends_on`	→	Convexity implies weak lower semicontinuity	declared	appendices/A-long-proofs.tex:22139
`depends_on`	→	The direct method	declared	appendices/A-long-proofs.tex:22139
`proves`	←	app:A-long-proofs@proof-273	declared	appendices/A-long-proofs.tex:22560