lemma A.233 Construction of the system

open in the book · appendices/A-long-proofs.tex:11692 · p. 2905

Rests on

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 definition 6.5 Open cover ¶
    - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on theorem 12.18 Projection theorem ¶
      - proves proof ch:10-hilbert-spaces@proof-11 ¶
    - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      - depends_on definition 5.41 Adjoint ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on theorem 12.46 Riesz representation ¶
      - proves proof ch:10-hilbert-spaces@proof-20 ¶
  - 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 definition 6.24 Metric ¶
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on lemma 6.30 Lebesgue number ¶
      - depends_on definition 6.26 Open ball; metric topology ¶
      - depends_on definition 6.5 Open cover ¶ ↺
      - depends_on definition 6.29 Sequential compactness ¶ ↺
      - proves proof ch:04-topology@proof-10 ¶
    - 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 ¶ ↺
- proves proof app:A-long-proofs@proof-140 ¶
depends_on theorem 12.18 Projection theorem ¶ ↺
proves proof app:A-long-proofs@proof-142 ¶

Supports

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

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Attainment	declared	appendices/A-long-proofs.tex:11706
`depends_on`	→	Restriction to an invariant closed subspace	declared	appendices/A-long-proofs.tex:11706
`depends_on`	→	Projection theorem	declared	appendices/A-long-proofs.tex:11706
`depends_on`	←	The eigenvalues tend to zero, with finite multiplicity	declared	appendices/A-long-proofs.tex:11751
`proves`	←	app:A-long-proofs@proof-142	declared	appendices/A-long-proofs.tex:11710