I've found on an ancient book a structure of the kind described below and I do not even know if it is a commonly known structure nowadays. For me it doesn't match neither a direct sum nor a direct product.
Can someone tell me something about this? (available references will be appreciate)
The complex vector space of functions defined by the following base $\{\alpha_{ij}\}= \{ \eta_i \tau_j\}$ where $\{\eta_i\}$ and $\{\tau_j\}$ are respectively ortonormal bases of trigonometric polynomials. An element $\overrightarrow{v}$ of this space will be of the form: $\overrightarrow{v} = \sum_{i,j=1}^\infty \alpha_{ij}\eta_i\tau_j$. A linear operator (matrix) over this space will be an infinite dimensional matrix with four indexes which I'm not sure if has anything to do with tensors.
If you require $\sum_{i,j} |\alpha_{ij}|^2 < \infty$, the vector space is the Hilbert space tensor product $\mathcal H = \mathcal H_\eta \otimes \mathcal H_\tau$ of the Hilbert space $\mathcal H_\eta$ generated by the $\eta_i$ and $\mathcal H_\tau$ generated by the $\tau_j$.