Since [S.sub.[omega]](R) is a Frechet space
, we have from Banach-Steinhaus theorem that T is linear and continuous; that is, T [member of] [S'.sub.[omega]](R).
Given Frechet space
R([F.sub.1],...,[F.sub.n]) of all n-dimensional random vectors X = ([X.sub.1], ..., [X.sub.n]), we have [F.sub.1], ..., [F.sub.n] as marginal distributions, where [F.sub.i](x) := P([X.sub.i] [less than or equal to] x), and the joint distribution function is [mathematical expression not reproducible], we have the following inequality:
As [C.sub.k](Y) is a Frechet space
, the Open Mapping Theorem [11, Theorem 14.4.6] implies that T : [C.sub.k](Y) [right arrow] [C.sub.k](X) is open, and hence it is a quotient map.
Let X be a Frechet space
with a family of seminorms [mathematical expression not reproducible] such that
In order to define the "multiplicity" in this generality, we recall that, associated to a continuous representation [pi] of a Lie group on a Banach space H, a continuous representation [[pi].sup.[infinity]] is defined on the Frechet space
[H.sup.[infinity]] of [C.sup.[infinity]]-vectors of H.
We denote by [C.sup.[infinity]]([C.sup.n]) the Frechet space
of complex-valued functions on [C.sup.n] endowed with the compact-open topology, while [L.sup.2]([C.sup.n],dm) denotes the usual Hilbert space of square integrable complex-valued functions F(z) on [C.sup.n] with respect to the usual Lebesgue measure dm.
Therefore, by the Banach-Steinhaus theorem for Frechet space
, we deduce that [[PI].sub.k] is equally a continuous linear operator on E, and the theorem is established.
Another way to understand this definition is the following: we choose a pre-Hilbert norm on the Frechet space
If [LAMBDA] = [R.sub.>0] we can assume that all occurring limits are countable and so [[epsilon].sub.(M)] (U, [R.sup.s]) is a Frechet space
. Moreover [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a Silva space, i.e.
It is well known that, if X is a Banach space, then C([R.sub.+]; X) can be organized in a canonical way as a Frechet space
, i.e., as a complete metric space in which the corresponding topology is induced by a countable family of seminorms.
It is well known that A (D) is a Frechet space
. By [J.sub.[alpha]] we shall denote the integration operator in the space A(D) defined by the formula
In the microlocal analysis we deal with the space of symbols which are infinitely differentiable functions and make it into Frechet space
by means of seminorms .