Alternatively, it can be argued using the uniform boundedness principle.

The uniform boundedness principle yields a simple non-constructive proof of this fact.

For infinity, the result is a more or less trivial corollary of the uniform boundedness principle.

In mathematics, the uniform boundedness principle or Banach-Steinhaus theorem is one of the fundamental results in functional analysis.

The least restrictive setting for the uniform boundedness principle is a barrelled space where the following generalized version of the theorem holds :

The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.

The uniform boundedness principle (also known as Banach-Steinhaus theorem) applies to sets of operators with uniform bounds.

The uniform boundedness principle in functional analysis provides sufficient conditions for uniform boundedness of a family of operators.

BCT1 is used in functional analysis to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle.

Using the uniform boundedness principle, one can show that the Fourier series, "typically", does not converge pointwise for elements in C(T).

Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in L(T) and the Banach-Steinhaus uniform boundedness principle.

Hahn's contributions to mathematics include the Hahn-Banach theorem and (independently of Banach and Steinhaus) the uniform boundedness principle.

Therefore by the uniform boundedness principle, for any x in T, the set of continuous functions whose Fourier series diverges at x is dense in C(T).

The reader is assumed to have a good background in undergraduate real and complex analysis, point set topology and elementary general functional analysis (Hahn-Banach theorem, uniform boundedness principle, Riesz-Kakutani theorem etc.).

By a transference result, the and problems are equivalent to one another, and as such, by an argument using the uniform boundedness principle, for any particular , norm convergence follows in both cases for exactly those where is the symbol of an bounded Fourier multiplier operator.