Fréchet Space
A Fréchet space is a complete and metrizable space, sometimes also with the restriction that the space be locally convex. The topology of a Fréchet space is defined by a countable family of seminorms. For example, the space of smooth functions on [0,1] is a Fréchet space. Its topology is the C-infty topology, which is given by the countable family of seminorms,
| ||f||_alpha=sup|D^alphaf|. |
Because f_n->f in this topology implies that f is smooth, i.e.,
| D^alphaf_n->D^alphaf, |
any Cauchy sequence has a limit in the space of smooth functions, i.e., it is a complete vector space.
See also
T1-SpacePortions of this entry contributed by Todd Rowland
Portions of this entry contributed by Margherita Barile
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References
Schaefer, H. H. Topological Vector Spaces. New York: Macmillan, 1966.Referenced on Wolfram|Alpha
Fréchet SpaceCite this as:
Barile, Margherita; Rowland, Todd; and Weisstein, Eric W. "Fréchet Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FrechetSpace.html