theorem A.456 Convexity in $q$ implies weak lower semicontinuity

Rests on

• depends_on equation 16.62 eq:calcvar-convexity-tangent ¶
• 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 A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
  • depends_on lemma A.454 Arzelà–Ascoli on an interval ¶
    • depends_on theorem 7.7 Bolzano–Weierstrass ¶
      • depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
        • depends_on definition A.36 Cut ¶
        • depends_on theorem A.40 Least-upper-bound property ¶
        • proves proof app:A-long-proofs@proof-32 ¶
      • depends_on corollary A.46 Monotone convergence ¶
        • depends_on theorem A.40 Least-upper-bound property ¶ ↺
        • proves proof app:A-long-proofs@proof-33 ¶
      • proves proof ch:05-real-analysis@proof-4 ¶
    • depends_on theorem 7.8 Cauchy criterion ¶
      • depends_on corollary A.47 Cauchy completeness ¶
        • depends_on corollary A.46 Monotone convergence ¶ ↺
        • depends_on theorem A.40 Least-upper-bound property ¶ ↺
        • proves proof app:A-long-proofs@proof-34 ¶
      • depends_on definition 7.4 Convergence ¶
        • depends_on definition 7.2 Absolute value ¶
      • proves proof ch:05-real-analysis@proof-5 ¶
    • proves proof app:A-long-proofs@proof-270 ¶
  • depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
    • depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
      • depends_on definition 10.85 Sobolev space ¶
        • depends_on definition 12.2 Hilbert space ¶ ↺
        • depends_on definition 10.83 Weak solution ¶
      • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on theorem 7.40 Continuous functions are integrable ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • proves proof ch:05-real-analysis@proof-24 ¶
    • depends_on lemma A.452 Young and Hölder ¶
      • depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶ ↺
      • depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
        • depends_on proposition 7.30 Leibniz rule ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-22 ¶
      • proves proof app:A-long-proofs@proof-268 ¶
    • proves proof app:A-long-proofs@proof-269 ¶
  • proves proof app:A-long-proofs@proof-271 ¶
• proves proof app:A-long-proofs@proof-272 ¶

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