theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$

open in the book · appendices/A-long-proofs.tex:23179 · p. 3024

Rests on

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 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 lemma 9.56 Lagrange identity ¶
      - 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 ¶
      - proves proof app:A-long-proofs@proof-276 ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - 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-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 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 ¶
- proves proof app:A-long-proofs@proof-279 ¶
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.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 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-280 ¶

Supports

depends_on lemma A.472 The pairing identity ¶

Neighborhood

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	$T$ is compact	declared	appendices/A-long-proofs.tex:23190
`depends_on`	→	The Green operator inverts $L_{K}$	declared	appendices/A-long-proofs.tex:23190
`depends_on`	→	Hilbert–Schmidt: compact self-adjoint operators	declared	appendices/A-long-proofs.tex:23190
`depends_on`	←	The pairing identity	declared	appendices/A-long-proofs.tex:23265
`proves`	←	app:A-long-proofs@proof-280	declared	appendices/A-long-proofs.tex:23194