lemma A.472 The pairing identity

Rests on

• depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
  • depends_on definition A.467 Green function of the shifted problem ¶
    • depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
      • depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
        • depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
        • depends_on definition A.438 Momentum variable and the accessory system ¶
        • proves proof app:A-long-proofs@proof-261 ¶
      • depends_on lemma 9.56 Lagrange identity ¶
        • depends_on definition 9.54 Sturm–Liouville differential operator ¶
        • proves proof ch:07-odes-sturm-liouville@proof-28 ¶
      • proves proof app:A-long-proofs@proof-264 ¶
    • depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
      • depends_on equation A.764 eq:app-sl-completeness-operator ¶
      • 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 lemma A.452 Young and Hölder ¶
        • proves proof app:A-long-proofs@proof-274 ¶
      • proves proof app:A-long-proofs@proof-276 ¶
  • depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶ ↺
  • depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶ ↺
  • proves proof app:A-long-proofs@proof-278 ¶
• depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
  • depends_on theorem A.470 $T$ is compact ¶
    • depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      • depends_on definition A.467 Green function of the shifted problem ¶ ↺
      • depends_on theorem 7.35 Mean value theorem ¶
        • depends_on proposition 7.29 Linearity ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-19 ¶
      • proves proof app:A-long-proofs@proof-277 ¶
    • 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 corollary A.46 Monotone convergence ¶
        • proves proof ch:05-real-analysis@proof-4 ¶
      • depends_on theorem 7.8 Cauchy criterion ¶
        • depends_on corollary A.47 Cauchy completeness ¶
        • depends_on definition 7.4 Convergence ¶
        • proves proof ch:05-real-analysis@proof-5 ¶
      • proves proof app:A-long-proofs@proof-270 ¶
    • proves proof app:A-long-proofs@proof-279 ¶
  • depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶ ↺
  • 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 proposition 12.21 Characterization of orthogonal projections ¶
        • depends_on definition 12.20 Orthogonal projection operator ¶
        • 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 proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
        • depends_on theorem 12.46 Riesz representation ¶
        • 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 ¶
        • 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 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 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 ¶
        • 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-280 ¶
• proves proof app:A-long-proofs@proof-281 ¶

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