The Lebesgue Space of Square-Integrable Functions on the Torus is Not Finite-Dimensional

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the initial segments $[n]$, the integer lattice $\mathbb{Z}^{n}$ and the real Hilbert space $L^{2}(\mathbb{T}^{n})$ are the ones fixed there.

Then $L^{2}(\mathbb{T}^{n})$, which is a vector space over $\mathbb{R}$ by Real Inner Product Spaces, is not finite-dimensional.