Imagine you investigate a set of objects $\mathcal{E}$, and you just realize this that $\mathcal{E}$ possesses an affine structure with respect to some real vector space $\mathcal{V}$ having a scalar product $\langle \cdot , \cdot \rangle$ (namely an euclidian affine space). My first thought was "Nice, I can equip $\mathcal{E}$ with a structure of metric space and explore the meaning of the associated convergence". But is there moreover some generic advantage to have an euclidian structure ?

One may formulate alternatively : "What informations you get for $\mathcal{E}$ knowing that there exists a notion of orthogonality on ?"

This is a very broad question, but this is motivated by a question I posted few months ago : What do we actually know about logarithmic energy ?

I guess I'm looking for an answer pointing some Theorems characterizing euclidian affine spaces between metric affine spaces, but also for your first reaction, or direction for investigation, after such a discovery.

Maybe it is of interest to precise that $\mathcal{E}$ is in fact a subspace of signed measures on $\mathbb{C}$ having total mass equals to $1$, and thus is an infinite dimensional real space.

EDIT : I precise my question,

Is there some geometric theory developed for infinite dimensional real pre-Hilbert spaces (which are not complete) ?

I emphasis that the elements of the space I have in mind are measures, and the scalar product has the form $$ \langle \mu, \nu \rangle = \iint K(x,y)d\mu(x)d\mu(y) $$ with a kernel $K : \mathbb{C}^2\rightarrow \mathbb{R}$ given by $$ K(x,y)=\log|x-y|. $$ Maybe there also exists some references for general kernels which are less singular ?