lemma A.250 Decomposition into cyclic subspaces

open in the book · appendices/A-long-proofs.tex:12470 · p. 2913

Rests on

depends_on definition A.248 Cyclic vector and cyclic subspace ¶
- depends_on proposition A.243 Continuous functional calculus ¶
  - depends_on proposition A.241 The polynomial calculus is isometric ¶
    - depends_on lemma A.240 Spectral mapping for polynomials ¶
      - depends_on definition 12.49 Resolvent set; spectrum ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - proves proof app:A-long-proofs@proof-146 ¶
    - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on definition 12.49 Resolvent set; spectrum ¶ ↺
      - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
      - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
      - proves proof app:A-long-proofs@proof-145 ¶
    - 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 app:A-long-proofs@proof-147 ¶
  - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
    - depends_on definition 12.60 Functional calculus ¶
      - depends_on definition 12.58 Projection-valued measure ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-32 ¶
  - depends_on theorem A.242 Stone–Weierstrass; quoted ¶
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - proves proof app:A-long-proofs@proof-148 ¶
- depends_on theorem 12.18 Projection theorem ¶
  - 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 5.24 Orthogonal vectors ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - proves proof ch:10-hilbert-spaces@proof-2 ¶
    - depends_on definition 12.16 Orthogonal complement ¶
      - depends_on definition 5.24 Orthogonal vectors ¶ ↺
      - depends_on equation 5.52 eq:lin-orthogonal-complement ¶
    - proves proof ch:10-hilbert-spaces@proof-8 ¶
  - depends_on theorem 12.14 Closest point in a closed convex set ¶
    - depends_on definition 12.13 Convex set ¶
      - depends_on definition 5.7 Vector subspace ¶
    - 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 ¶
    - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - depends_on equation 5.44 eq:lin-norm-assoc ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-3 ¶
    - proves proof ch:10-hilbert-spaces@proof-7 ¶
  - proves proof ch:10-hilbert-spaces@proof-9 ¶
depends_on definition 12.32 Separable Hilbert space ¶
- depends_on definition 12.2 Hilbert space ¶ ↺
- depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶ ↺
depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
- depends_on definition 12.86 Internal orthogonal decomposition ¶
  - depends_on definition 12.29 Orthonormal basis ¶
    - depends_on definition 12.16 Orthogonal complement ¶ ↺
    - 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 definition 12.16 Orthogonal complement ¶ ↺
- depends_on proposition 12.85 The direct sum is a Hilbert space ¶
  - depends_on definition 12.84 External direct sum ¶
    - depends_on definition 5.101 External direct sum ¶
      - depends_on definition 4.33 Vector space ¶
      - depends_on definition 5.5 Linear combination ¶
    - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
      - depends_on definition 12.2 Hilbert space ¶ ↺
      - depends_on definition 5.18 Inner product ¶ ↺
  - depends_on proposition 12.10 $\ell^{2}$ is complete ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on example 12.9 The sequence space $\ell^{2}$ ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-5 ¶
  - proves proof ch:10-hilbert-spaces@proof-41 ¶
- 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@proof-42 ¶
proves proof app:A-long-proofs@proof-155 ¶

Supports

depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
- depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Cyclic vector and cyclic subspace	declared	appendices/A-long-proofs.tex:12478
`depends_on`	→	Separable Hilbert space	declared	appendices/A-long-proofs.tex:12478
`depends_on`	→	Expansion in an orthogonal decomposition	declared	appendices/A-long-proofs.tex:12478
`depends_on`	←	Direct-integral form of the spectral theorem	declared	appendices/A-long-proofs.tex:13962
`proves`	←	app:A-long-proofs@proof-155	declared	appendices/A-long-proofs.tex:12482