Examples of dual-spaces

There are three main classes of duality.

1. Hilbert spaces are self-dual i.e. \(X = X^*\). The geometric view is that each functional corresponds directly to an element in \(X\)

Example 13

In \(\mathbb{R}^n\) each functional arises via the inner product, thus \((\mathbb{R}^n)^* = \mathbb{R}^n)\). Similarly, for \(\mathbb{C}^n\).

Example 14

In \(L^2(\Omega)\) every functional is given by

\[ f[x] = \int x(t) g(t) \mathrm{d}t,\quad\mbox{for some}\ g \in L^2(\Omega). \]

2. Reflexive spaces are those for which \(X^{**} = X\).

Example 15

The spaces \(L^p(\Omega)\) for \(1 < p < \infty\) are reflexive with \((L^p)^* = L^q\) where \(\frac{1}{p} + \frac{1}{q} = 1\).

3. Non-reflexive spaces are those where \(X \subset X^{**}\).

Example 16

\(L^\infty(\Omega)\) 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.