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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:

  1. What is the motivation of constructing dual space of a normed space?

  2. What is the main role of dual space of a normed space in functional analysis?

Thank you for all your construction and comments.

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3 Answers

up vote 8 down vote accepted

When we consider finite dimensional spaces $V$, we do this usually by choosing a basis $v_1, \ldots, v_n$ and look at the hereby given isomorphism $T\colon V \to \mathbb K^n$ with $Tv_i = e_i$. Another way to see this is that we are describing a general $v \in V$ by its coordinates $\lambda_i(v) := \pi_i(Tv)$ where $\pi_i \colon \mathbb K^n \to \mathbb K$ is the projection onto the $i$-th factor. It's helpful to have a unique description by real numbers at hand, for we can use all properties of $\mathbb K$ we have already established. The main point here is that knowing all $\lambda_i(v)$ gives us $v$.

In infinite dimensions, generally we have no basis to hand. AC gives us the existence, but this is not very helpful in computations. The idea of considering functionals may be seen as a generalization of the $\lambda_i \colon V \to \mathbb K$ from above. Instead of looking at some (well choosen) functionals from $V$ to $\mathbb K$ we look at all of them, that is, at the set $X^* = \{x^* \colon V \to \mathbb K \mid x^* \text{ linear, continuous}\}$ and find that it works quite well as a replacement of the coordinate functionals.

For example, Hahn-Banach helps us to distiguish elements of $X$ in the following sense: $$ x \ne y \iff \exists x^* : x^*(x) \ne x^*(y). $$ As with the coordinates, our generialized "coordinates" $\bigl(x^*(x)\bigr)$ for $x\in X$ allow us to reformulate problems in $X$ to problems in $\mathbb K$, which are often easier to solve.

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Thank you for your answer. –  blindman Oct 24 '12 at 8:54
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Suppose you are studying some finite dimensional topological vector space $X$ (which, in turn, must be isomorphic to $\mathbb{C}^d$), then I guess the most natural thing to do is to introduce coordinates. Or, equivalently, choose a basis $\{e_j\}_{j=1}^d$, so that each $x=\sum x_j e_j\in X$ is represented by $(x_1,x_2,\dots, x_d)$. The mappings $x\mapsto x_j$ and their linear spans are exactly continuous linear functionals for $X$, and for $X=\mathbb{C}^d$,the dual space is also $\mathbb{C}^d$.So you can see the dual space for infinite dimensional spaces is just a generalization of coordinates.

However, continuous linear functionals are much more useful than coordinates, mainly because the topological structure and algebraic structure are not so well-behaved for infinite dimensional spaces (they determine each other for finite dimensional spaces). They become indispensable throught the collection of theorems bearing the name of Hahn-Banach, with which (at least in norm space setting) one can separate points from closed subspaces, from convex bodies, etc.

So I think from here it is already clear continuous linear functionals are like coordinates, but they can somehow blend both algebraic (linear) structure and the topology. When the space is complicated, study the functions over the space (that respect certain properties of the space). This is like the theme behind dual spaces.

From another point of view, they are also very natural in the sense that so many objects (evaluation, integration and distribution) can be realised as continuous over certain spaces.

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Thank you for your answer. –  blindman Oct 24 '12 at 8:55
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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.

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Thank you for your answer. –  blindman Oct 24 '12 at 8:56
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