X-Nico

Banach space

It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed.

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C(X)

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If one of the two spaces X or Y is complete (or reflexive, separable, etc.) then so is the other space.

Initial value problem

More generally, the unknown function y can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions.

Baire category theorem

Note that neither of these statements implies the other, since there are complete metric spaces which are not locally compact (the irrational numbers with the metric defined below; also, any Banach space of infinite dimension), and there are locally compact Hausdorff spaces which are not metrizable (for instance, any uncountable product of non-trivial compact Hausdorff spaces is such; also, several function spaces used in Functional Analysis; the uncountable Fort space).

Dunford–Pettis property

In functional analysis, the Dunford–Pettis property, named after Nelson Dunford and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous.

Radon–Riesz property

Although it would appear that Johann Radon was one of the first to make significant use of this property in 1913, M. I. Kadets and V. L. Klee also used versions of the Radon–Riesz property to make advancements in Banach space theory in the late 1920s.

see also

Komura

Kōmura's theorem, result on the differentiability of absolutely continuous Banach space-valued functions