After reading this question on MathOverflow I began to wonder to what extent Fourier transforms can be defined on Hilbert spaces, Banach spaces or their duals with the weak-* topology (or even a general TVS). In the infinite-dimensional setting we can lose local compactness and so classical harmonic analysis is inadequate.
My guess is that for a sufficiently nice topological vector space $E$, it's Pontryagin dual $\hat E$ (i.e. it's space of characters) is simply the topological dual $E^*$. I also think that for a nice enough measure $\mu$ over $E$ we can define the Fourier transform as $$ \hat\mu: E^*\to\mathbb{C} \\ \hat\mu(\varphi) = \int_E e^{i\langle \varphi, x\rangle} \,\mathrm{d}\mu(x) $$ where $\langle\cdot,\cdot\rangle:E^*\times E\to\mathbb{R}$ is the duality pairing. Furthermore, I would expect $\hat\mu=0$ iff $\mu=0$.
I'm particularly interested in the case where $E$ is some dual Banach space with the weak-* topology. For example, $E$ being $M(K)$ with the weak-* topology for some compact Hausdorff $K$ (in this case I'd be looking at Fourier analysis for measures on measures).
Is my intuition above correct? Are there any reference to study the Fourier theory of measures on such infinite-dimensional spaces? Or if the theory falls apart, that would be good to know as well. Thanks!