Finite norm of sequence implies convergence of sequence?
Consider an $L^p(X)$ space, for example a probability space. Suppose that I have a sequence of function $x_n\in X$, such that $\lim_{n\to\infty} \lVert x_n\rVert_p<\infty$. Does this imply that $\{x_n\}_{n}^\infty$ converge to an $x\in X$ under the same norm? If not what are the typical conditions to suffice this?
Solution 1:
It does not. A simple example is to look at the standard basis $\{e_n \}_{n \in \mathbb{N}}$ for $\ell^2(\mathbb{N})$. Then $$\lVert e_n\rVert_2 = 1 $$ but the sequence does not converge to anything.