Internal direct sum of infinitely many subspaces I understand what the (external) direct product of an infinite number of vector spaces. I understand the internal direct sum of a finite number of vector spaces. I think I understand what an external direct sum of vector spaces is.
My question is, 
Does it make sense to talk about an internal direct sum of infinitely many subspaces of a vector space? If so, how is this defined?
 A: Yes. If $V$ is any vector space and $(W_i)_{i\in I}$ is a family of subspaces of $V$, we write $$V =\bigoplus_{i\in I}W_i$$if the following conditions hold:
(i) for every $v\in V$ there are $i_1,\ldots,i_k\in I$ ($k$ depends on $v$) and $w_{i_1}\in W_{i_1}$, ..., $w_{i_k}\in W_{i_k}$ such that $$v=w_{i_1}+\cdots + w_{i_k}.$$This is written as $V=\sum_{i\in I}W_i$.
(ii) for every $j\in I$, we have $W_j\cap \sum_{i\neq j}W_i = \{0\}$.
Condition (ii) ensures that the decomposition of $v$ in (i) is unique. For example, if $(e_i)_{i\in I}$ is a basis for $V$ over a field $\mathbb{K}$, then $$V=\bigoplus_{i\in I}\mathbb{K}e_i,$$where each $\mathbb{K}e_i$ is the line spanned by $e_i$.
A: Let me add a more categorical description to Ivo Terek's answer. Given a family of subspaces $\{W_i\}_{i\in I}$ of $V$, we always have a canonical map from the "external" direct sum
$$ \bigoplus_{i \in I} W_i \to V,$$
given by the universal property of direct sum: a map out of the direct sum is a collection of maps out of the summands. We say that $V$ is the internal direct sum of the family $\{W_i\}_{i \in I}$ if this canonical map is an isomorphism. Spelling out the conditions for this map to be an isomorphism give exactly the conditions in Ivo Terek's answer.
