Difference between revisions of "Fréchet space"
Ulf Rehmann (talk | contribs) m (moved Frechet space to Fréchet space over redirect: accented title) |
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− | + | {{MSC|46A04}} | |
+ | {{TEX|done}} | ||
− | + | A Fréchet space is a complete metrizable locally convex topological vector space. Banach spaces furnish examples of Fréchet spaces, but several important function spaces are Fréchet spaces without being Banach spaces. Among these are: the Schwartz space $\mathscr{S}(\R^n)$ of all infinitely-differentiable complex-valued functions on $\R^n$ that decrease at infinity, as do all their derivatives, faster than any polynomial, with the topology given by the system of semi-norms | |
− | + | \[ | |
− | where | + | p_{\alpha,\beta}(x) = \sup_{t \in \R^n} |
+ | \left| | ||
+ | t_1^{\beta_1} \cdots t_n^{\beta_n} | ||
+ | \frac{ | ||
+ | \partial^{\alpha_1 + \cdots + \alpha_n} x(t_1,\ldots,t_n) | ||
+ | }{ | ||
+ | \partial t_1^{\alpha_1} \cdots \partial t_n^{\alpha_n} | ||
+ | } | ||
+ | \right|, | ||
+ | \] | ||
+ | where $\alpha$ and $\beta$ are non-negative integer vectors; the space $H(D)$ of all holomorphic functions on some open subset $D$ of the complex plane with the topology of uniform convergence on compact subsets of $D$, etc. | ||
A closed subspace of a Fréchet space is a Fréchet space; so is a quotient space of a Fréchet space by a closed subspace; a Fréchet space is a [[Barrelled space|barrelled space]], and therefore the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]] is true for mappings from a Fréchet space into a locally convex space. If a separable locally convex space is the image of a Fréchet space under an open mapping, then it is itself a Fréchet space. A one-to-one continuous linear mapping from a Fréchet space onto a Fréchet space is an isomorphism (an analogue of a theorem of Banach). | A closed subspace of a Fréchet space is a Fréchet space; so is a quotient space of a Fréchet space by a closed subspace; a Fréchet space is a [[Barrelled space|barrelled space]], and therefore the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]] is true for mappings from a Fréchet space into a locally convex space. If a separable locally convex space is the image of a Fréchet space under an open mapping, then it is itself a Fréchet space. A one-to-one continuous linear mapping from a Fréchet space onto a Fréchet space is an isomorphism (an analogue of a theorem of Banach). | ||
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Fréchet spaces are so named in honour of M. Fréchet. | Fréchet spaces are so named in honour of M. Fréchet. | ||
− | ====References==== | + | ====References==== |
− | + | {| | |
− | + | |- | |
− | + | |valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Topological vector spaces", Springer (1987) (Translated from French) | |
− | + | |- | |
− | + | |valign="top"|{{Ref|KeNa}}||valign="top"| J.L. Kelley, I. Namioka, "Linear topological spaces", Springer (1963) | |
− | + | |- | |
− | + | |valign="top"|{{Ref|Kö}}||valign="top"| G. Köthe, "Topological vector spaces", '''1''', Springer (1969) | |
− | + | |- | |
− | + | |valign="top"|{{Ref|RoRo}}||valign="top"| A.P. Robertson, W.S. Robertson, "Topological vector spaces", Cambridge Univ. Press (1973) | |
+ | |- | ||
+ | |valign="top"|{{Ref|Sc}}||valign="top"| H.H. Schaefer, "Topological vector spaces", Macmillan (1966) | ||
+ | |- | ||
+ | |} |
Latest revision as of 00:13, 27 July 2012
2010 Mathematics Subject Classification: Primary: 46A04 [MSN][ZBL]
A Fréchet space is a complete metrizable locally convex topological vector space. Banach spaces furnish examples of Fréchet spaces, but several important function spaces are Fréchet spaces without being Banach spaces. Among these are: the Schwartz space $\mathscr{S}(\R^n)$ of all infinitely-differentiable complex-valued functions on $\R^n$ that decrease at infinity, as do all their derivatives, faster than any polynomial, with the topology given by the system of semi-norms \[ p_{\alpha,\beta}(x) = \sup_{t \in \R^n} \left| t_1^{\beta_1} \cdots t_n^{\beta_n} \frac{ \partial^{\alpha_1 + \cdots + \alpha_n} x(t_1,\ldots,t_n) }{ \partial t_1^{\alpha_1} \cdots \partial t_n^{\alpha_n} } \right|, \] where $\alpha$ and $\beta$ are non-negative integer vectors; the space $H(D)$ of all holomorphic functions on some open subset $D$ of the complex plane with the topology of uniform convergence on compact subsets of $D$, etc.
A closed subspace of a Fréchet space is a Fréchet space; so is a quotient space of a Fréchet space by a closed subspace; a Fréchet space is a barrelled space, and therefore the Banach–Steinhaus theorem is true for mappings from a Fréchet space into a locally convex space. If a separable locally convex space is the image of a Fréchet space under an open mapping, then it is itself a Fréchet space. A one-to-one continuous linear mapping from a Fréchet space onto a Fréchet space is an isomorphism (an analogue of a theorem of Banach).
Fréchet spaces are so named in honour of M. Fréchet.
References
[Bo] | N. Bourbaki, "Topological vector spaces", Springer (1987) (Translated from French) |
[KeNa] | J.L. Kelley, I. Namioka, "Linear topological spaces", Springer (1963) |
[Kö] | G. Köthe, "Topological vector spaces", 1, Springer (1969) |
[RoRo] | A.P. Robertson, W.S. Robertson, "Topological vector spaces", Cambridge Univ. Press (1973) |
[Sc] | H.H. Schaefer, "Topological vector spaces", Macmillan (1966) |
Fréchet space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_space&oldid=27212