lemma A.249 The cyclic case

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 proposition A.245 The measures $\mu_{x,y}$ ¶
  • depends_on proposition A.243 Continuous functional calculus ¶ ↺
  • 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 5.37 Linear transformation ¶
        • depends_on definition 5.19 Norm ¶
      • depends_on definition 5.46 Functional ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
    • depends_on theorem 12.18 Projection theorem ¶ ↺
    • proves proof ch:10-hilbert-spaces@proof-24 ¶
  • proves proof app:A-long-proofs@proof-149 ¶
• depends_on theorem A.244 Riesz–Markov; quoted ¶ ↺
• proves proof app:A-long-proofs@proof-154 ¶

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