TITLE: Basis in infinite dimensional Hilbert spaces
QUESTION [5 upvotes]: 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

REPLY [8 votes]: In Hilbert space theory and Functional analysis the term 'basis' has a different definition than in linear algebra. For this reason one often encounters the term 'Hamel basis' for the latter and 'Hilbert space basis' in the first case. 
In the Hilbert space case the requirement is that each vector can be represented as in infinite sum, equality being understood in the sense of convergence in the Hilbert space norm. The term 'Schauder basis' is also used in B spaces for which infinite sums are allowed.
See e.g. http://en.wikipedia.org/wiki/Schauder_basis