lemma A.234 The eigenvalues tend to zero, with finite multiplicity

Rests on

• depends_on lemma A.233 Construction of the system ¶
  • depends_on lemma A.232 Attainment ¶
    • depends_on lemma A.230 Sequential characterisation ¶
      • depends_on definition 12.41 The operator classes ¶
        • depends_on definition 6.9 Compact set ¶
        • depends_on proposition 12.21 Characterization of orthogonal projections ¶
        • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      • depends_on theorem 6.31 Compactness and sequential compactness ¶
        • depends_on definition 6.9 Compact set ¶ ↺
        • depends_on definition 6.29 Sequential compactness ¶
        • depends_on lemma 6.30 Lebesgue number ¶
        • proves proof ch:04-topology@proof-11 ¶
      • proves proof app:A-long-proofs@proof-139 ¶
    • depends_on proposition 12.42 Elementary consequences ¶
      • depends_on definition 12.41 The operator classes ¶ ↺
      • 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 ch:10-hilbert-spaces@proof-22 ¶
    • 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-141 ¶
  • depends_on lemma A.231 Restriction to an invariant closed subspace ¶
    • depends_on definition 12.41 The operator classes ¶ ↺
    • depends_on lemma A.230 Sequential characterisation ¶ ↺
    • 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 12.16 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 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 app:A-long-proofs@proof-140 ¶
  • depends_on theorem 12.18 Projection theorem ¶ ↺
  • proves proof app:A-long-proofs@proof-142 ¶
• depends_on lemma A.230 Sequential characterisation ¶ ↺
• proves proof app:A-long-proofs@proof-143 ¶

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