theorem 12.33 Every separable Hilbert space is $\ell^{2}$

Rests on

• depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
  • depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    • depends_on definition 6.24 Metric ¶
      • depends_on definition 3.43 Map ¶
        • depends_on definition 3.24 Quantifiers ¶
        • depends_on definition 3.28 Set ¶
      • depends_on definition 3.38 Ordered pair and Cartesian 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 4.33 Vector space ¶
      • 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 ¶
  • proves proof ch:10-hilbert-spaces@proof-13 ¶
• depends_on proposition 12.10 $\ell^{2}$ is complete ¶
  • depends_on axiom 7.1 Completeness of $\R$ ¶
  • depends_on example 12.9 The sequence space $\ell^{2}$ ¶
    • 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 5.18 Inner product ¶ ↺
  • proves proof ch:10-hilbert-spaces@proof-5 ¶
• 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 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 12.18 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 ¶
        • depends_on definition 12.13 Convex set ¶
        • depends_on definition 12.2 Hilbert space ¶ ↺
        • depends_on proposition 12.6 Parallelogram law and polarization ¶
        • proves proof ch:10-hilbert-spaces@proof-7 ¶
      • proves proof ch:10-hilbert-spaces@proof-9 ¶
    • 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-17 ¶

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