Consider a sequence of linear regression problems: $$ Y^{(n)}_n = X^{(n)}_{n \times p } \beta_{p}^{(n)} $$ where $p$ is fixed and $X^{(n)}_{n \times p }$ is full rank for all $n$.
My question is: What is the general topology (Banach space, locally convex space, etc) in which we can prove the convergence of $\beta$ parameters? $$ \beta^{(n)} = (X^{{(n)}\top} X^{(n)})^{-1} X^{{(n)}\top} Y^{(n)}. $$ Thanks in advance.