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A bounded linear operator between normed spaces
Definition
Let and be normed spaces over the same scalar field , read in the real case from A norm on a real vector space, the induced metric, and the dictionary with the metric axioms and in the complex case from Real and complex scalar conventions for normed spaces. A linear map (Linear map between vector spaces over the same field) is a bounded linear operator when there is a real constant such that
Any such is called a bound for .
Remarks
- The zero operator is bounded with bound .
- A bound is not unique: if works and , then works as well.
- When , every linear map is bounded with bound .
Depends on
Used by
- A bounded operator that is bounded below Definition
- A complemented closed subspace of a normed space Definition
- A topological isomorphism of normed spaces Definition
- Bounded left inverses and bounded right inverses Definition
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum Definition
- The spaces (B(X,Y)) and (B(X)) of bounded linear operators Definition
- Coordinate projections and inclusions on a finite product Banach space Example
- Differentiation on polynomials is unbounded for the supremum norm Example
- Forward and backward shifts on classical sequence spaces and their exact operator norms Example
- The evaluation functional on (C(K)) has norm one Example
- The quotient by the kernel is isometric to the range with its induced quotient norm Example
- A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm Theorem
- A bounded operator that vanishes on a subspace factors uniquely through the normed quotient Theorem
- A closed subspace is complemented exactly when it is the range of a bounded projection Theorem
- For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent Theorem
- Under Dependent Choice, a surjective bounded operator between Banach spaces has a bounded right inverse exactly when its kernel is complemented Theorem
- Under Dependent Choice, an injective bounded operator between Banach spaces has a bounded left inverse exactly when its range is closed and complemented Theorem
Dependency tree · two levels
18 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)