I am reading a paper on functional data analysis and they use a Hilbert space as living space for their functions. Then they defined a "dictionary" $\mathcal{D}\subset\mathcal{H}$ where the elements (functions) of this set are noted $d$, then they define the inner product $\langle , \rangle_{\mathcal{H}}$ on this Hilbert space. Here is my problem :
When they consider $d\in\mathcal{D}$ and $f\in\mathcal{H}$ and they look at $\langle f, d\rangle_{\mathcal{H}}$, they say that this is the projection of $f$ on $\mathcal{D}$ but I don't see why.
From my knowledge, this inner product can at best describes the coordinates of the projection of $f$ on $\mathcal{D}$ and it requires that there exists an orthonormal basis on this subspace, something that is not mentionned in the paper.
Did I miss some properties of infinite dimensional vector space ?