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Attempts to find classes of locally convex topological vector spaces on which the uniform boundedness principle holds eventually led to barrelled spaces. That is, the least restrictive setting for the uniform boundedness principle is a barrelled space, where the following generalized version of the theorem holds (Bourbaki 1987, Theorem III.2.1):
The Baire category theorem (BCT) is an important result in general topology and functional analysis.The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense).
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. ... A special case of this is the uniform boundedness principle ...
Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in L 1 (T) and the Banach–Steinhaus uniform boundedness principle. As typical for existence arguments invoking the Baire category theorem, this proof is nonconstructive.
The Baire Characterisation Theorem states that a real valued function f defined on a Banach space X is a Baire-1 function if and only if for every non-empty closed subset K of X, the restriction of f to K has a point of continuity relative to the topology of K.
In particular, a Vitali set does not have the property of Baire. [4] Already weaker versions of choice are sufficient: the Boolean prime ideal theorem implies that there is a nonprincipal ultrafilter on the set of natural numbers; each such ultrafilter induces, via binary representations of reals, a set of reals without the Baire property. [5]
René-Louis Baire (French:; 21 January 1874 – 5 July 1932) was a French mathematician most famous for his Baire category theorem, which helped to generalize and prove future theorems. His theory was published originally in his dissertation Sur les fonctions de variables réelles ("On the Functions of Real Variables") in 1899.
Every uniformly convergent sequence of bounded functions is uniformly bounded.; The family of functions () = defined for real with traveling through the integers, is uniformly bounded by 1.