theorem A.461 Completeness in the weighted and in the energy norm

Rests on

• 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 equation 16.66 eq:calcvar-sl-problem ¶
• depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
  • depends_on definition 12.41 The operator classes ¶
    • 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 ¶
    • depends_on proposition 12.21 Characterization of orthogonal projections ¶
      • depends_on definition 12.20 Orthogonal projection operator ¶
        • depends_on theorem 12.18 Projection theorem ¶
      • depends_on theorem 12.18 Projection theorem ¶ ↺
      • proves proof ch:10-hilbert-spaces@proof-11 ¶
    • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      • depends_on definition 5.41 Adjoint ¶
        • depends_on definition 5.18 Inner product ¶
        • depends_on definition 5.37 Linear transformation ¶
      • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
        • depends_on definition 12.35 Bounded operator; operator norm ¶
        • depends_on proposition 12.8 Absolutely convergent series test ¶
        • proves proof ch:10-hilbert-spaces@proof-19 ¶
      • depends_on theorem 12.46 Riesz representation ¶
        • depends_on definition 12.45 Continuous linear functional; the dual ¶
        • depends_on theorem 12.18 Projection theorem ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-24 ¶
      • proves proof ch:10-hilbert-spaces@proof-20 ¶
  • depends_on proposition 12.43 Norm of a self-adjoint operator ¶
    • depends_on proposition 12.42 Elementary consequences ¶
      • depends_on definition 12.41 The operator classes ¶ ↺
      • depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
        • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
        • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-21 ¶
      • proves proof ch:10-hilbert-spaces@proof-22 ¶
    • depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
    • proves proof ch:10-hilbert-spaces@proof-23 ¶
  • depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    • depends_on corollary 12.19 Double complement; the density criterion ¶
      • depends_on proposition 12.17 The complement is always a closed subspace ¶
        • depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
        • depends_on definition 12.16 Orthogonal complement ¶
        • proves proof ch:10-hilbert-spaces@proof-8 ¶
      • depends_on theorem 12.18 Projection theorem ¶ ↺
      • proves proof ch:10-hilbert-spaces@proof-10 ¶
    • depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
      • depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶
        • depends_on definition 5.27 Orthonormal basis ¶
        • depends_on equation 5.49 eq:lin-orthonormality ¶
      • depends_on theorem 12.18 Projection theorem ¶ ↺
      • proves proof ch:10-hilbert-spaces@proof-14 ¶
    • depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
      • depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶ ↺
      • depends_on definition 12.2 Hilbert space ¶
        • depends_on definition 5.18 Inner product ¶ ↺
        • depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
        • depends_on equation 5.44 eq:lin-norm-assoc ¶
      • proves proof ch:10-hilbert-spaces@proof-15 ¶
    • proves proof ch:10-hilbert-spaces@proof-16 ¶
  • proves proof ch:10-hilbert-spaces@prooflink-1 ¶
• proves proof app:A-long-proofs@proof-282 ¶

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