Download Characterizations of Inner Product Spaces (Operator Theory by Amir PDF

By Amir

ISBN-10: 3764317744

ISBN-13: 9783764317744

Each mathematician operating in Banaeh spaee geometry or Approximation concept is aware, from his personal experienee, that almost all "natural" geometrie houses might faH to carry in a generalnormed spaee except the spaee is an internal produet spaee. To reeall the weIl recognized definitions, this implies IIx eleven = *, the place is an internal (or: scalar) product on E, Le. a functionality from ExE to the underlying (real or eomplex) box enjoyable: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =

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2]). By a (cochain) complex of topological vector spaces L' we understand any sequence of topological vector spaces L' (i E Z) together with continuous linear mappings d,: L' ~ £1+1 satisfying d'+1 0 d, = 0 for all i, With each such complex we associate the graduated topological vector space L' = {L'} endowed with the product topology, and the differential d: L' ~ L' of degree 1 defined by means of the equality dx = d,x for x E L'. 12 For the space Jf'(L') to be separated it is necessary and sufficient that B'(L') should be closed in L', Proof.

Since (d~E) Mt}d/) = 0, there exists a differential operator M2 of type p2 - t E2 such that (d~E) Ml ) = M2 d~F). By analogy, since (d~F) MIl )d~E) = 0, there exists a differential operator Mil of type E2 - t p2 such that (d~F) MIl) = Mil d~E). Further we consider a differential operator F ) of type pI - t pI given by F ) = 1 - M l- l Ml - d~F) h~F). 5 that F) d~F) = O. So there exists a differential operator h~F) of type p2 - t pI such that F ) = h~F) d~F). By analogy there exists a differential operator h~E) of type E2 - t El such that 1 - Ml M l- l - d~E) h~E) = h~E) d~E).

6) in degree z, endowed with the natural topologies, will he denoted respectively by Z'(q:(E )) ker P: q:(E') -+ q:(E,+I), B'(q:(E )) imP: q:(E,-I) -+ q:(E') and H'(

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