Question

Dual space definition in real analysis with example

Answer 1

Dual space (functional analysis)



In mathematics, particularly in the branch offunctional analysis, a dual space refers to the space of all continuous linear functionals on a real or complex Banach space. The dual space of a Banach space is again a Banach space when it is endowed with the operator norm. If Xis a Banach space then its dual space is often denoted by X'.

Contents

1 Definition2 The bidual space and reflexive Banach spaces3 Dual pairings4 References5 Further reading

[edit]Definition

Let X be a Banach space over a field F which is real or complex, then the dual space X' of  is the vector space over F of all continuous linear functionals  when F is endowed with the standard Euclidean topology.

The dual space  is again a Banach space when it is endowed with the operator norm. Here the operator norm  of an element  is defined as:



where  denotes the norm on X.

please mark as brainliest please please