theorem 16.81 Douglas; Radó

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:2566 · p. 660

Rests on

depends_on equation 16.7 eq:calcvar-minimal-surface ¶
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@prooflink-4 ¶

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type	direction	node	provenance	where
`depends_on`	→	eq:calcvar-minimal-surface	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2573
`depends_on`	→	The direct method	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2573
`proves`	←	ch:14-calculus-of-variations@prooflink-4	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2594