Uniform boundedness principle

The uniform boundedness principle states that a pointwise bounded family of continuous linear operators on a Banach space has uniformly bounded operator norms.

Baire category theorem: A complete metric space cannot be a countable union of nowhere-dense sets. Completeness supplies the category argument that converts pointwise bounds into a bound on a ball.
Banach–Steinhaus theorem: Another name for the uniform boundedness principle. This theorem is also widely known by this name, especially in functional analysis.
Pointwise convergence: A sequence of functions converges pointwise when it converges at each input separately. Pointwise behavior alone does not generally provide uniform bounds on the functions.
Contrapositive of the uniform boundedness principle: If operator norms in a family are unbounded, some vector has unbounded images under that family. This form turns norm blow-up into a witness vector.
Banach space: A complete normed vector space. Completeness is the essential hypothesis behind the theorem's Baire-category proof.
Fourier series: A representation of a periodic function as a sum of sine and cosine waves. Uniform boundedness helps show that convergence of Fourier partial sums cannot hold everywhere for every continuous function.
Uniform convergence: A sequence of functions converges uniformly when one error bound works simultaneously at every input. Uniform convergence is stronger than pointwise control, while the principle derives a different kind of uniformity.
Unbounded linear operator: A linear map that is not bounded with respect to the specified norms. The principle constrains families of bounded operators; its conclusion fails if individual operators lack boundedness.
Continuous linear operator: A linear map between normed vector spaces that is continuous, equivalently bounded. The theorem applies to families of these maps and bounds their operator norms.
Hahn–Banach theorem: A theorem extending bounded linear functionals from a subspace while preserving their norm. Together, these theorems provide foundational tools for reasoning about bounded linear functionals.

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