Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Normed subspace
Definition
Let be a normed space and let be a linear subspace in the sense of Linear subspace of a vector space. The normed subspace is the vector space equipped with the restricted norm
Its induced metric is the restriction of the ambient norm metric to , so the inclusion is an isometric embedding in the sense of Isometry, isometric embedding, and the subspace metric on a subset.
Remarks
- No new notation is forced: once the ambient norm is fixed, the restricted norm is usually written with the same symbol.
- Completeness and closedness for a normed subspace are therefore comparison statements between the restricted metric and the ambient one, not new definitions.
Depends on
Used by
- A finite-dimensional normed subspace is closed Corollary
- Restriction to a closed invariant subspace is a C0-semigroup and its generator is the part Corollary
- A complete domain is necessary for sequential uniform boundedness Counterexample
- A complemented closed subspace of a normed space Definition
- Infinitesimal generator of a C0-semigroup Definition
- The positive-type Gaussian on the real line and its cyclic model Example
- A closed subspace of a Banach space is Banach Lemma
- A complete normed subspace is closed under countable choice Lemma
- Riesz lemma Lemma
- A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm Theorem
- A bounded real linear functional on a subspace of a real normed space extends with the same norm Theorem
- Geometric Hahn--Banach theorem for subspaces Theorem
- Hardy–Littlewood–Sobolev fractional integration inequality Theorem
- Well-posedness of the abstract Cauchy problem is equivalent to generation Theorem
Dependency tree · two levels
20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
- Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)