proposition 12.96 Operators on a tensor product

Rests on

• 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 definition 12.35 Bounded operator; operator norm ¶
      • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on definition 5.19 Norm ¶
        • depends_on definition 4.33 Vector space ¶ ↺
    • depends_on proposition 12.8 Absolutely convergent series test ¶
      • depends_on definition 5.19 Norm ¶ ↺
      • depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
        • depends_on definition 6.24 Metric ¶
      • proves proof ch:10-hilbert-spaces@proof-4 ¶
    • proves proof ch:10-hilbert-spaces@proof-19 ¶
  • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    • depends_on definition 5.41 Adjoint ¶
      • depends_on definition 5.18 Inner product ¶
        • depends_on definition 4.33 Vector space ¶ ↺
      • depends_on definition 5.37 Linear transformation ¶ ↺
    • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
    • 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 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 ¶
      • proves proof ch:10-hilbert-spaces@proof-24 ¶
    • proves proof ch:10-hilbert-spaces@proof-20 ¶
  • proves proof ch:10-hilbert-spaces@proof-21 ¶
• depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
  • depends_on definition 12.94 Tensor product of Hilbert spaces ¶
    • 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 5.110 Bilinear map ¶
      • depends_on definition 5.37 Linear transformation ¶ ↺
    • depends_on definition 5.18 Inner product ¶ ↺
  • depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
    • depends_on definition 5.27 Orthonormal basis ¶
      • depends_on definition 5.26 Orthogonal basis ¶
        • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.24 Orthogonal vectors ¶
      • depends_on definition 5.25 Unit vector ¶
        • depends_on definition 5.19 Norm ¶ ↺
    • depends_on proposition 5.28 Gram–Schmidt ¶
      • depends_on definition 5.14 Linear independence ¶
        • depends_on definition 5.5 Linear combination ¶
      • depends_on definition 5.18 Inner product ¶ ↺
      • depends_on definition 5.27 Orthonormal basis ¶ ↺
      • depends_on equation 5.44 eq:lin-norm-assoc ¶ ↺
      • proves proof ch:03-linear-algebra-representations@proof-5 ¶
    • proves proof ch:10-hilbert-spaces@proof-12 ¶
  • 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 definition 5.27 Orthonormal basis ¶ ↺
        • depends_on equation 5.49 eq:lin-orthonormality ¶
      • 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-45 ¶
• proves proof ch:10-hilbert-spaces@proof-46 ¶

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