Consider a function $f: [-1,1] \rightarrow \mathbb{R}$ expanded in terms of Legendre functions: $$ f(x) = \sum_{n=0}^{\infty} a_n P_n(x) $$ Is it possible to define vectors with infinitely many elements: \begin{align} \mathbf{a} &= \left( a_0, a_1, \dots \right) \\ \mathbf{P}(x) &= \left( P_0(x), P_1(x), \dots \right) \end{align} and say $$ f(x) = \mathbf{a} \cdot \mathbf{P}(x) $$ ?
What about a matrix $\mathbf{P}$ whose columns correspond to $P_0, P_1, \dots$, and whose rows are defined for all $x \in [-1,1]$, so that row "x" of $\mathbf{P}$ would be $\mathbf{P}(x)$. In case could we say $$ \mathbf{f} = \mathbf{P} \mathbf{a} $$ where $\mathbf{f}$ is a vector of all uncountably infinitely many $f(x)$ for $x \in [-1,1]$ and we use the normal matrix-vector multiplication above? Is it possible to define such "infinite" vectors and "infinite" matrices and do normal linear algebra operations with them?
Is there a branch of mathematics which studies these sorts of things?