proposition A.245 The measures $\mu_{x,y}$

open in the book · appendices/A-long-proofs.tex:12178 · p. 2910

Rests on

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 definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 5.47 Inverse of a 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 ¶
    - 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 definition 12.49 Resolvent set; spectrum ¶ ↺
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-27 ¶
    - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
      - depends_on proposition 12.42 Elementary consequences ¶
      - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
      - proves proof ch:10-hilbert-spaces@proof-23 ¶
    - proves proof app:A-long-proofs@proof-145 ¶
  - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
  - 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 definition 12.41 The operator classes ¶
      - depends_on proposition 12.21 Characterization of orthogonal projections ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - depends_on definition 12.58 Projection-valued measure ¶ ↺
      - depends_on proposition 12.43 Norm of a self-adjoint operator ¶ ↺
      - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
      - proves proof ch:10-hilbert-spaces@prooflink-2 ¶
  - 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 ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- proves proof app:A-long-proofs@proof-148 ¶
depends_on theorem A.244 Riesz–Markov; quoted ¶
- depends_on definition 6.9 Compact set ¶ ↺
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 definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
- 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 ¶
- proves proof ch:10-hilbert-spaces@proof-24 ¶
proves proof app:A-long-proofs@proof-149 ¶

Supports

depends_on lemma A.249 The cyclic case ¶
depends_on proposition A.246 Bounded Borel functional calculus ¶
- depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
  - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
    - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
- depends_on proposition A.261 Spectral theorem for a unitary operator ¶
  - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Continuous functional calculus	declared	appendices/A-long-proofs.tex:12191
`depends_on`	→	Riesz–Markov; quoted	declared	appendices/A-long-proofs.tex:12191
`depends_on`	→	Riesz representation	declared	appendices/A-long-proofs.tex:12191
`depends_on`	←	The cyclic case	declared	appendices/A-long-proofs.tex:12441
`depends_on`	←	Bounded Borel functional calculus	declared	appendices/A-long-proofs.tex:12237
`proves`	←	app:A-long-proofs@proof-149	declared	appendices/A-long-proofs.tex:12195