Examples of dual-spaces - Applied Functional Analysis

There are three main classes of duality.

Hilbert spaces are self-dual i.e. $X = X^*$ . The geometric view is that each functional corresponds directly to an element in $X$ .
Example 1
In $\mathbb{R}^n$ each functional arises via the inner product, thus $(\mathbb{R}^n)^* = \mathbb{R}^n$ . Similarly, for $\mathbb{C}^n$ .
Example 2
In $L^2(\Omega)$ every functional is given by
$f[x] = \int x(t) g(t) \mathrm{d}t,\quad\mbox{for some}\ g \in L^2(\Omega).$
(1)
Reflexive spaces are those for which $X^{**} = X$ .
Example 3
The spaces $L^p(\Omega)$ for $1 < p < \infty$ are reflexive with $(L^p)^* = L^q$ where $\frac{1}{p} + \frac{1}{q} = 1$ .
Non-reflexive spaces are those where $X \subset X^{**}$ .
Example 4
$L^\infty(\Omega)$ is not separable i.e. it has too many independent directions. From the previous result we would guess that $(L^\infty)^* = L^1$ however we know that functions in $L^1$ correspond to regular measures on $\Omega$ . But this won’t let us distinguish / measure functions in $L^\infty$ , recall that the topology i.e. norm on $L^\infty$ is very discrening. Thus we need singular measure function properties on sets of measure zero.