[Math] Basis in infinite dimensional Hilbert spaces

Question

Let $H$ be a Hilbert space with a countable basis $B$. Does it mean that any vector $x\in H$ can be expressed as a finite linear combination of elements from $x$, or as an infinite linear combination?

Thanks in advance

Answer 1

Take $\ell^2$ for example, i.e. the square-summable sequences of complex numbers with inner product $$\langle x,y\rangle = \sum_{n=1}^\infty x_n\overline{y_n}. $$ This has the countable orthonormal basis $$\{(1,0,0,\ldots), (0,1,0,\ldots), (0,0,1,0,\ldots),\ldots\}. $$ As $$\sum_{n=1}^\infty 2^{-n} = 1<\infty, $$ we have $$(2^{-1}, 2^{-2}, 2^{-3},\ldots)\in\ell^2, $$ and it is clear that this element cannot be written as a finite linear combination of the basis elements.