lemma A.464 $H_{E}$ is a Hilbert space

Rests on

• depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
  • depends_on definition A.462 The weighted space and the energy space ¶
    • 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 definition 16.75 Rayleigh quotient ¶
      • depends_on definition 9.54 Sturm–Liouville differential operator ¶
      • depends_on equation 16.42 eq:calcvar-sl-from-isoperimetric ¶
  • 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 definition 7.12 Product of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.27 Differentiable implies continuous ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-14 ¶
      • depends_on theorem 7.34 Rolle ¶
        • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on theorem 7.24 Extreme value theorem ¶ ↺
        • proves proof ch:05-real-analysis@proof-18 ¶
      • proves proof ch:05-real-analysis@proof-22 ¶
    • proves proof app:A-long-proofs@proof-268 ¶
  • proves proof app:A-long-proofs@proof-274 ¶
• depends_on theorem 12.12 Riesz–Fischer ¶
  • depends_on example 12.11 The function space $L^{2}$ ¶
    • 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 proposition 12.8 Absolutely convergent series test ¶
    • depends_on definition 5.19 Norm ¶
      • depends_on definition 4.33 Vector space ¶ ↺
    • 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 ¶
    • proves proof ch:10-hilbert-spaces@proof-4 ¶
  • proves proof ch:10-hilbert-spaces@proof-6 ¶
• proves proof app:A-long-proofs@proof-275 ¶

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