The space $\mathcal{L}^p(\mathbb{R}^n)$ of functions $f$ such that $\int |f(x)|^p\, d^nx$ converges is only a seminormed rather than a normed vector space, because any function $f$ whose support has Lebesgue measure zero has $||f||_p := \int |f(x)|^p\, d^nx = 0$, even if $f$ is not identically zero. In order to turn it into a normed vector space $L^p(\mathbb{R}^n)$, we need to mod out by the kernel of the $p$-norm, which is the set of functions whose support has Lebesgue measure zero (or equivalently, we need to identify functions that agree almost everywhere).
This raises the natural question of whether this quotient space $L^p(\mathbb{R}^n)$ has a natural section; that is, whether for every equivalence class $[f] \in L^p(\mathbb{R}^n)$ there is a natural canonical representative square-integrable function $f \in \mathcal{L}^p(\mathbb{R}^n)$. Is there generally a natural section of $\mathcal{L}^p(\mathbb{R}^n)$ space? (I'm using the word "natural" loosely, not in any kind of mathematically precise sense.)
For the Hilbert space $L^2(\mathbb{R}^n)$ with a given orthonormal basis $\{\phi_n(x)\}$, there does seem to be a natural canonical representative $f_c \in [f]$, given by the generalized Fourier series for $[f]$: $$f_c(x) := \sum_{n=0}^\infty \langle f, \phi_n \rangle \phi_n, \qquad \qquad \langle f, \phi_n \rangle := \int_{\mathbb{R}^n} f(x)\, \phi_n(x)\, d^nx.$$ Does this choice of section depend on the choice of basis $\{ \phi_n \}$? (Obviously it would be "nicer" if it didn't.)