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Dual spaces

Can someone dumb it down a little what is a dual space of the Banach space V?

It's (usually defined as) the vector space of all continuous (a.k.a. bounded) linear functionals on V. So if I say

f \in V^*

, then f is a function

V \to \mathbb{R}

(or

\mathbb{C}

if you are working with a complex Banach space) which is continuous (or bounded, under the operator norm) and linear.

Very rarely, the dual space is defined generically as the space of all linear functionals on V - including the discontinuous / unbounded ones. This is valid for any vector space V, whereas the above definition requires a normed vector space.