theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1150 · p. 425

Rests on

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 definition 6.1 Topological space ¶
    - 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 proposition 12.17 The complement is always a closed subspace ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - proves proof ch:10-hilbert-spaces@proof-9 ¶
  - 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 4.33 Vector space ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
    - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.19 Norm ¶
    - depends_on proposition 12.8 Absolutely convergent series test ¶
      - depends_on definition 5.19 Norm ¶ ↺
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - proves proof ch:10-hilbert-spaces@proof-4 ¶
    - 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 definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.46 Functional ¶
    - 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 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 definition 5.26 Orthogonal basis ¶
      - depends_on definition 5.25 Unit vector ¶
    - 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 ¶

Supports

depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
- depends_on lemma A.472 The pairing identity ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The operator classes	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1163
`depends_on`	→	Norm of a self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1163
`depends_on`	→	Completeness, expansion, Parseval	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1163
`depends_on`	←	Completeness in the weighted and in the energy norm	declared	appendices/A-long-proofs.tex:22801
`depends_on`	←	Spectral decomposition and completeness in $L^{2}_{r}$	declared	appendices/A-long-proofs.tex:23190
`proves`	←	ch:10-hilbert-spaces@prooflink-1	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1179