Currently reading through Optimization by Vector Space methods, specifically chapter 5 Dual Spaces.
I've read several times about this topic, but there's a specific insight here that I've never thought of, which I quote.
Dual space plays a role analogous to the inner product in Hilbert Space; by suitable interpretation we can develop results extending the projection theorem solution of minimum norm problems to arbitrary normed linear spaces.
The analogy in this book is given in the context of optimisation. But I wonder if something some other analogies can be extrapolated.
As examples I am thinking the following.
Series expansion. Say you have an Hilbert Space $X$ with a orthonormal basis $\left\{\phi_j\right\}_{j\in\mathbb{Z}}$ then any vector $x \in X$ can be written uniquely as
$$ x = \sum_{j\in\mathbb{Z}} \left\langle x, \phi_j \right\rangle \phi_j $$
Can these expansions be generalized using Dual Spaces whenever the space is not a Hilbert one?
I would be thrilled to learn about more about these analogies.