proposition 12.36 Boundedness is continuity

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:897 · p. 423

Rests on

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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 5.19 Norm ¶
  - depends_on definition 4.33 Vector space ¶ ↺
depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
- depends_on definition 6.6 Continuous map ¶
  - depends_on definition 3.53 Preimage ¶
  - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on definition 6.1 Topological space ¶ ↺
- depends_on definition 6.26 Open ball; metric topology ¶
  - 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 6.2 Open set ¶ ↺
  - depends_on definition 6.1 Topological space ¶ ↺
- proves proof ch:04-topology@proof-9 ¶
proves proof ch:10-hilbert-spaces@proof-18 ¶

Supports

depends_on theorem 12.74 Hellinger–Toeplitz ¶
- depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶

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`	→	Bounded operator; operator norm	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:902
`depends_on`	→	$\varepsilon$–$\delta$ characterization	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:902
`depends_on`	←	Hellinger–Toeplitz	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2028
`proves`	←	ch:10-hilbert-spaces@proof-18	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:905