lemma A.230 Sequential characterisation

open in the book · appendices/A-long-proofs.tex:11568 · p. 2904

Rests on

depends_on definition 12.41 The operator classes ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- depends_on proposition 12.21 Characterization of orthogonal projections ¶
  - depends_on definition 12.20 Orthogonal projection operator ¶
    - 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 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 definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 5.19 Norm ¶
    - depends_on proposition 12.8 Absolutely convergent series test ¶
      - depends_on definition 5.19 Norm ¶ ↺
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - proves proof ch:10-hilbert-spaces@proof-4 ¶
    - proves proof ch:10-hilbert-spaces@proof-19 ¶
  - 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 ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-24 ¶
  - 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 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 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 6.24 Metric ¶ ↺
    - depends_on definition 6.2 Open set ¶ ↺
    - depends_on definition 6.1 Topological space ¶ ↺
  - 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 ¶

Supports

depends_on lemma A.232 Attainment ¶
- depends_on lemma A.233 Construction of the system ¶
  - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
depends_on lemma A.231 Restriction to an invariant closed subspace ¶
- depends_on lemma A.233 Construction of the system ¶ ↺
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`	→	The operator classes	declared	appendices/A-long-proofs.tex:11573
`depends_on`	→	Compactness and sequential compactness	declared	appendices/A-long-proofs.tex:11573
`depends_on`	←	Attainment	declared	appendices/A-long-proofs.tex:11632
`depends_on`	←	Restriction to an invariant closed subspace	declared	appendices/A-long-proofs.tex:11601
`depends_on`	←	The eigenvalues tend to zero, with finite multiplicity	declared	appendices/A-long-proofs.tex:11751
`proves`	←	app:A-long-proofs@proof-139	declared	appendices/A-long-proofs.tex:11576