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For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent
Statement
Let and be normed spaces over the same scalar field, and let be linear. Then the following are equivalent:
- is bounded.
- is continuous at .
- is continuous on .
- is Lipschitz.
Facts & Assumptions
Given: Normed spaces and , a linear map , a real , and a vector .
A bounded linear operator has a constant with for every (A bounded linear operator between normed spaces).
A map is Lipschitz when one constant controls all distances, and every Lipschitz map between metric spaces is continuous (Lipschitz map, -Hölder map for rational , and contraction, Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent).
Continuity at a point in a metric space is the - condition of Continuity of a map between metric spaces, at a point and globally, in the - form.
Proof
Assume is bounded, with constant from [L1]. Then for all , so is Lipschitz.
If is continuous on , then in particular it is continuous at , so .
Assume is continuous at . Applying [L3] with gives such that implies .
Step 1.1 proves , and [L2] gives .
Let with and put . Then , so by step 1.3. By linearity, , hence . The same inequality is trivial at , so is bounded.
Thus . Combining steps 2.1, 1.2, and 2.2 gives all four equivalences.
Depends on
- A bounded linear operator between normed spaces
- Lipschitz map, $\alpha$-Hölder map for rational $0 < \alpha \le 1$, and contraction
- Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent
- Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
Used by
- Volterra operator is Hilbert Schmidt and quasinilpotent Example
- A compact remainder estimate forces closed range Lemma
- Compositions with a compact operator are compact Lemma
- Kernel of identity minus compact is finite dimensional Lemma
- Maximal orthogonal family of cyclic reducing subspaces Lemma
- Norm point of a compact self adjoint operator is an eigenvalue up to sign Lemma
- Orthogonal complement of an eigenspace is invariant Lemma
- Positive square root of a compact positive operator Lemma
- Range of identity minus compact is closed Lemma
- Restriction and extension estimates are dual Lemma
- Successive approximation turns a closure-ball inclusion into an actual preimage Lemma
- A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm Theorem
- Bounded inverse theorem Theorem
- Hardy–Littlewood–Sobolev fractional integration inequality Theorem
- Hilbert–Schmidt operators are compact Theorem
- Singular value decomposition for compact operators Theorem
- Spectral theorem for compact self adjoint operators Theorem
- Weak Neumann solvability on the mean-zero subspace Theorem
Dependency tree · two levels
22 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)