definition A.248 Cyclic vector and cyclic subspace

open in the book · appendices/A-long-proofs.tex:12422 · p. 2913

ground object -- no derivation owed

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 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 definition 5.18 Inner product ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - proves proof ch:10-hilbert-spaces@proof-1 ¶
    - 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 5.5 Linear combination ¶
  - depends_on definition 12.2 Hilbert space ¶ ↺
  - 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 ¶

Supports

depends_on lemma A.249 The cyclic case ¶
depends_on lemma A.250 Decomposition into cyclic subspaces ¶
- 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`	→	Continuous functional calculus	declared	appendices/A-long-proofs.tex:12430
`depends_on`	→	Projection theorem	declared	appendices/A-long-proofs.tex:12430
`depends_on`	←	The cyclic case	declared	appendices/A-long-proofs.tex:12441
`depends_on`	←	Decomposition into cyclic subspaces	declared	appendices/A-long-proofs.tex:12478