Why do we need dual space

Question

In functional analysis there are many places where dual space is mentioned, but I still don't understand the real power of that concept. Why do we need the dual space?

Answer 1

More generally, if $X$ and $Y$ are sets (resp. vector spaces over a common field $k$), we can consider the set $Y^X$ of all mappings (resp. the vector space $\mathrm{Hom}(X,Y)$ of all linear mappings) from $X$ to $Y$. These are obviously interesting to study if you think in terms of structures and structure preserving maps; and dual spaces are simply a special case (namely $\mathrm{Hom}(X,k)$). Also $\mathrm{Hom}$ behaves very nicely with respect to short exact sequences, and is deeply related to the tensor product.

Now apparently you are interested in a functional analysis point of view: here we assume that the dual space is the topological dual space (i.e. continuous linear functionals); the dual space is always complete (regardless of whether $X$ is complete or not) and points can be separated by elements of the dual space, which is often useful. A dual space which is particularly interesting is $\mathscr{D}'$: the space of distributions (here $\mathscr{D}$ is $C^\infty_0$ endowed with its canonical LF topology); it can be shown that if $P(\partial)$ is any linear constant coefficient partial differential operator, then there is a distribution $E$ (called a fundamental solution for $P(\partial)$) such that $P(\partial)E=\delta_0$ in the sense of distributions (this is the theorem of Malgrange-Ehrenpreis). This is a very important result because of the following corollary: if $f\in C^\infty_0$, then the PDE $P(\partial)u=f$ admits a $C^\infty$ solution explicitly given by $u=E\ast f$. In physics (say electrodynamics) this is used all the time to solve PDE's. Also, one often seeks weak solutions for the PDE $P(\partial)u=f$ (i.e. this equation is satisfied in the distribution sense) and then tries to prove that they are strong (usual) solutions; this is a very convenient method.

Answer 2

The dual space is a concept that shows up in greater generality in Linear Algebra. If $V$ is a vector space then its dual space $V^*$ is the set of all linear functionals from $V$ to its base field $F$.

When $V$ is a finite dimensional vector space, then so is $V^*$ and $\text{dim } V^*=\text{dim }V$. Thus there is a linear isomorphism between them when they are taken over the same field. Why is this important? Such an isomorphism gives us an example of a unnatural isomorphism in category theory. Since $V^*$ is also a vector space, it also has a dual denoted $V^{**}$ called the double-dual of $V$. Unlike between $V$ and $V^*$ there is a natural isomorphism between $V$ and $V^{**}$.