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Uniform boundedness principle
Statement
Assume DC. Let be a Banach space, a normed space, and let be a family of bounded linear operators (A bounded linear operator between normed spaces). If for every , then , where the norm is The operator norm as the least bound and as the unit-sphere or unit-ball supremum.
Facts & Assumptions
Given: DC, as in the statement, and pointwise boundedness.
Proof
Put . Each is closed (an intersection of inverse images of closed balls), and pointwise boundedness gives .
For , both and lie in that ball, so for every .
Rescaling a nonzero to yields ; the same is clear for . Thus for every .
Depends on
Used by
- A pointwise limit of bounded operators is bounded with the liminf norm bound Corollary
- A separately continuous bilinear map on Banach spaces is jointly continuous Corollary
- A continuous function with divergent Fourier series at a prescribed point Counterexample
- The Carleson maximal operator is not strong type (1,1) Counterexample
- A semigroup with continuity at zero is uniformly bounded on every compact time interval Lemma
- Countable compactness closes in the bidual Lemma
- Negative partial-homogeneity levels trap all sufficiently close forcings Lemma
- Partial-homogeneity flows preserve box separation Lemma
- Signed narrow convergence controls variation tails and Bochner moments Lemma
- Baire dichotomy for a pointwise-defined family of bounded linear operators Theorem
- Compact operator sends weakly convergent sequences to norm convergent sequences Theorem
- Lax equivalence for a common Banach-space scheme Theorem
Dependency tree · two levels
10 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
- Buhler--Salamon, Functional Analysis, Theorem 2.1 (standard reference, not scraped)