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I am dealing with the dual spaces for the first time.

I just wanted to ask is their any practical application of Dual space or is it just some random mathematical thing? If there is, please give a few.

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In Lax's linear algebra book, the dual space is used to give a very elegant and simple proof of the rank-nullity theorem. – littleO Aug 11 '14 at 11:07
Do you mean applications in mathematics, or in other areas of science? – Federico Poloni Aug 11 '14 at 12:11
I was actually looking for its applications in science and technology. – Manish Aug 11 '14 at 17:33
In technology, I'd answer "almost no application", since everything can be done using $\mathbb{R}^n$ (or $\mathbb{C}^n$ for fancier applications) and transposes. In physics, dual spaces are very useful when formalizing mechanics and introducing quantum mechanics. But basically the sentence "Physicists use X in quantum mechanics or string theory" is true if you replace X with any mathematical object. :) – Federico Poloni Aug 12 '14 at 18:20

Distributions, or "generalized functions", such as the Dirac delta are most conveniently described by studying the dual space of a space of "nice" (smooth or something of the kind) functions. Heuristic calculations with distributions are commonplace in physics, but if you need a rigorous understanding of what is going on and what operations are allowed, you end up studying dual spaces.

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Vector fields and differential forms are dual in this sense but behave quite differently. Thus, one can pull back differential forms by smooth maps but there is no analogous operation for vector fields.

A finite dimensional vector space is of course isomorphic to its dual, but the isomorphism is not canonical. In other words, there is no natural way of identifying them.

To get a feeling for what "natural" means in this context, you have to start thinking about global objects such as vector bundles. For example, the Mobius band can be thought of as a vector bundle over the circle, with fiber a line. If one could construct a natural isomorphism between a fiber and the line $\mathbb R$, this would lead to a trivialisation of the vector bundle and would imply that the Mobius band is actually a cylinder, which it is not.

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I know $V$ and $V^{**}$ are canonical isomorphism. But I've no idea what does it mean "canonical" and "natural way"? – Shuchang Aug 11 '14 at 10:39
Actually $V$ and $V^{**}$ are only guaranteed to be isomorphic if $V$ is finite-dimensional. – Mikhail Katz Aug 11 '14 at 10:41
Yes. What is "canonical" or "natural"? – Shuchang Aug 11 '14 at 10:43
I added a related comment to my answer. – Mikhail Katz Aug 11 '14 at 10:48

When dealing with finite-dimensional vector spaces, it makes not much sense to study dual spaces. Every finite-dimensional vector space over the fields of real or complex numbers can be equipped with a scalar product. There is then no need to introduce a dual space in finite-dimensional linear algebra imho.

The picture changes completely with infinite-dimensional spaces. These spaces allow (and call!) for a much richer theory.

Here is one example, where a functional (i.e. an element from the dual space) enters the room: Let $f:X \to \mathbb R$ be a directionally differentiable function. Then the directional derivative of $f$ at $x$ is a mapping that maps every direction $\delta x$ to a real number $f'(x;\delta x)$. If this mapping $\delta x\mapsto f'(x;\delta x)$ is linear with respect to the direction, it is an element of the dual space, $f'(x) \in X^*$.

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Locally convex spaces have a very rich theory for their dual spaces, and much of functional analysis (as the name suggests) is devoted to the study of such spaces. Consider how important the concept of a basis is in the finite dimensional setting and then note that the continuous linear functionals are a natural generalization to infinite dimensional spaces.

Of course, the prime example would be Hilbert spaces, where the inner product allows one to identify the dual space with the space itself in a very direct manner. In a sense this obscures the role of the dual space, but you should take note of how important the inner product (and thus the linear functionals) are for the theory.

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Dual spaces enter naturally if one considers the weak topology on the original space. Sometimes a differential equation has a weak solution but not a strong one (i.e. in the original=strong, topology). Well known examples of dual spaces are $L^p$ spaces. The dual of ${L^p}(R,dx)$, $1≤p<∞$ is ${L^q}(R,dx)$ where $p^{-1}+q^{-1}=1$. You can find many details in the volumes by Reed and Simon.

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