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By Ditkine V., Proudnikov A.

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But u" is an isometry: Imu" is dense if and only if uq is surjective. D. u: 10. The epimorphism theorem 29 Thus, surjectivity of every mapping uq is indeed a weaker property than the surjectivity of u. 1. - We say that the mapping u is presurjective if u verifies the following condition: (PS) For every equicontinuous subset A ofSpecE and every yeF, there is a regular section s over A such that -1 us = 'lI7B(Y) , B=u*(A). We recall that 'lI7B(Y) is the regular section over B defined by y ('1I7 is the analog of w when F is substituted for E; see p.

There is a linear functional x* on E, extending x~ and such that Ix*1 ~ V. This is one of the standard versions of the Hahn-Banach theorem and its proof will not be given here. Corollary 1. - Let V be a seminorm on E, x a point of E. There is a linear functional X* on E such that Ix*l~v and

Is injective and Imu. 1) is a neighborhood of 0 in F the preimage under u. of any equicontinuous set is equicontinuous when F is barrelled) (7) u is presurjective. 3), whence the equivalence of (5), (6), (7). l: we see that (5) implies (1). 1. - Let E and F be two Frichet spaces, u:E-F a continuous linear map. 1) are equivalent. 11. Criteria of presurjectivity In this section, we consider two locally convex TVS, E and F, and a continuous linear map u:E-F. 1). Later on, these conditions will be applied to some situations in Linear POE theory.

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Calcul operationnel by Ditkine V., Proudnikov A.


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Categories: Differential Equations