TITLE: The role of dual space of a normed space in functional analysis
QUESTION [9 upvotes]: We have known that dual space of a normed space is very important in functional analysis.
I would like to ask two questions related dual space of a normed space:

What is the motivation of constructing dual space of a normed space?
What is the main role of dual space of a normed space in functional analysis?

Thank you for all your construction and comments.

REPLY [3 votes]: Continuous linear functionals are "measurement devices" specifically designed to observe different aspects of vectors in your vector space. Looking at a dual space is a way of studying the original space by looking at all possible measurements you could make on it.
The functionals (measurements) are restricted to continuous linear functions since they are sufficient to completely determine vectors in your space. Adding in nonlinear or discontinuous functionals would be redundant and complicate things.
Furthermore, linear functionals are those measurement devices where the measured quantity can be ascribed "units" (length, mass, time, whatever) consistent with the units of the original space, and continuous functionals are measurement devices wherein small changes to the quantity produce small changes in the measurement.