Fréchet space | Wikipedia audio article

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00:01:00 1 Definitions
00:03:20 2 Constructing Fréchet spaces
00:04:23 3 Examples
00:04:59 4 Properties and further notions
00:07:08 5 Differentiation of functions
00:07:12 6 Fréchet manifolds and Lie groups
00:14:20 7 Generalizations
00:15:33 8 Remarks
00:19:25 9 Notes
00:20:32 10 See also
00:20:39 11 References
00:21:07 Remarks
00:21:17 Notes
00:21:26 See also



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SUMMARY
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In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). Fréchet spaces are locally convex spaces that are complete with respect to a translation-invariant metric. In contrast to Banach spaces, the metric need not arise from a norm.
Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the lack of a norm, many important results in functional analysis, like the open mapping theorem, the closed graph theorem, and the Banach–Steinhaus theorem, still hold.
Spaces of infinitely differentiable functions are typical examples of Fréchet spaces.