Handbook of Complex Analysis by Krantz Steven G.;

Author:Krantz, Steven G.;
Language: eng
Format: epub
Publisher: CRC Press LLC
Published: 2021-12-30T00:00:00+00:00

When this has to be interpreted as a limit. More explicitly one defines

In addition to the algebraic dual E⋆, which is defined for any vector space, we consider for any normed vector space E also its dual space, denoted by E′ and consisting of all continuous linear forms on E. These are the linear mappings such that for some constant C. On the dual we define the norm dual to by

(9.19)

It follows that for all and all .

When , we may identify both E⋆ and E′ with Rn, and the evaluation of ξ at the point x, i.e., the number , is then the inner product, defined by . The Euclidean norm , defined by , is dual to itself:

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