proposition A.282 The fibre maps are continuous on $\Phi$

Rests on

• depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
  • depends_on lemma A.250 Decomposition into cyclic subspaces ¶
    • 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 proposition 12.61 Uniqueness of the continuous functional calculus ¶
        • depends_on theorem A.242 Stone–Weierstrass; quoted ¶
        • 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 theorem 12.14 Closest point in a closed convex set ¶
        • 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 5.18 Inner product ¶
        • depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
        • depends_on equation 5.44 eq:lin-norm-assoc ¶
      • 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 proposition 12.85 The direct sum is a Hilbert space ¶
        • depends_on definition 12.84 External direct sum ¶
        • depends_on proposition 12.10 $\ell^{2}$ is complete ¶
        • 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 12.27 Best approximation and Bessel's inequality ¶
        • depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
        • proves proof ch:10-hilbert-spaces@proof-16 ¶
      • proves proof ch:10-hilbert-spaces@proof-42 ¶
    • proves proof app:A-long-proofs@proof-155 ¶
  • depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    • depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      • depends_on definition 12.72 Symmetric; self-adjoint ¶
        • depends_on definition 12.69 Operator with a domain ¶
        • depends_on definition 12.71 Adjoint of a densely defined operator ¶
      • depends_on proposition A.259 The generator is self-adjoint ¶
        • depends_on proposition A.258 The generator is closed ¶
        • depends_on proposition A.256 Density ¶
        • depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
        • proves proof app:A-long-proofs@proof-162 ¶
      • depends_on proposition 12.73 The adjoint is always closed ¶
        • depends_on definition 12.70 Graph; closed and closable operators ¶
        • depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-35 ¶
      • proves proof app:A-long-proofs@proof-163 ¶
    • depends_on proposition A.246 Bounded Borel functional calculus ¶
      • depends_on proposition A.243 Continuous functional calculus ¶ ↺
      • 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 theorem 12.46 Riesz representation ¶
        • proves proof app:A-long-proofs@proof-149 ¶
      • depends_on theorem 12.46 Riesz representation ¶ ↺
      • proves proof app:A-long-proofs@proof-150 ¶
    • depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      • depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺
      • 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 theorem A.238 Spectral theorem, both forms ¶
        • 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 app:A-long-proofs@proof-151 ¶
        • proves proof app:A-long-proofs@proof-153 ¶
        • proves proof app:A-long-proofs@proof-156 ¶
      • proves proof app:A-long-proofs@proof-164 ¶
    • proves proof app:A-long-proofs@proof-165 ¶
  • depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
  • proves proof app:A-long-proofs@proof-177 ¶
• depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
  • depends_on definition A.278 Countably Hilbert nuclear space ¶
    • depends_on definition 12.103 Gelfand triple ¶
      • depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
        • depends_on definition 12.45 Continuous linear functional; the dual ¶
        • depends_on theorem 12.46 Riesz representation ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-25 ¶
      • depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
      • depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
    • depends_on definition 12.2 Hilbert space ¶ ↺
• proves proof app:A-long-proofs@proof-178 ¶

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