I am studying basic functional analysis, and I'm trying to understand the difference between $L^p$ spaces defined on open vs. closed intervals, and convergence in such spaces.
For example, I have an exercise that that says: Suppose that $f_n \to f$ uniformly on $[0,1]$ and show that $f_n \to f$ in $L^2((0,1))$. I think the argument is simply that $$||f_n-f||_{L^2}^2 = \int_{0}^{1}|f_n(x)-f(x)|^2dx \leq \int_{0}^{1}(\sup_{t\in[0,1]}|f_n(t)-f(t)|)^2dx = (\sup_{t\in[0,1]}|f_n(t)-f(t)|)^2 \to 0, \ \mathrm{as} \ n \to \infty$$
However, why would we say that the convergence is in $L^2((0,1))$ rather than in $L^2([0,1])$?