Banach–Steinhaus theorem

The Banach–Steinhaus theorem states that a pointwise bounded family of continuous linear operators on a Banach space is uniformly bounded in operator norm.

Pointwise boundedness: A family of functions is pointwise bounded when its values at each fixed input remain bounded across the family. This is the theorem’s hypothesis: each input has a finite bound shared across all operators.
Strong operator topology: The topology on a space of operators in which convergence means pointwise convergence on every vector. Pointwise boundedness is naturally expressed through the operators’ action on individual vectors.
Uniform convergence: A sequence of functions converges uniformly when one error bound works simultaneously at every point of the domain. Uniform convergence controls function values directly, while this theorem derives uniform operator norms from pointwise bounds.
Stefan Banach: Stefan Banach was a Polish mathematician and a founder of modern functional analysis. The theorem bears Banach’s name and belongs to the theory he helped establish.
Uniform boundedness: A family of functions is uniformly bounded when one finite bound applies to every function and every input in a specified set. The theorem’s conclusion strengthens separate bounds at each input into one operator-norm bound.
Weak convergence: A sequence in a normed space converges weakly when every continuous linear functional sends it to a convergent scalar sequence. Banach–Steinhaus helps control operators defined through weakly convergent families.
Equicontinuity: A family of functions is equicontinuous when one continuity condition works uniformly across the family. Equicontinuity is a distinct family-wide condition; pointwise boundedness alone does not imply it.
Hugo Steinhaus: Hugo Steinhaus was a Polish mathematician whose work spanned analysis, probability, and geometry. The theorem’s alternate name honors Steinhaus alongside Banach.
Baire category theorem: In a complete metric space, a countable union of closed sets with empty interiors cannot cover the space. Completeness lets the proof show that some pointwise boundedness set contains an open ball.
Fourier partial sums: Finite sums of terms in a Fourier series, formed by retaining frequencies up to a chosen cutoff. Their norm growth reveals why pointwise convergence of sums need not imply uniform operator bounds.

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