Infinite system of vectors
We say that a vector is a linear combination of an infinite system of vectors , if there exists a finite subset of , such that is its linear combination. A system of vectors is linearly independent, if none of its vectors is a combination of rest of them. It is a basis of a space , if it is linearly independent and every vector of is its linear combination.
E.g., system , where is the infinite sequence of zeroes with one only on -th place is linearly independent in the space of all real sequences . But it is not a basis of this space, since the sequence of infinitely many ones is not a linear combination of this system (notice the finite subset part of the definition of linear combination). But it is a basis of space of all sequences which have zeroes from some moment on.
Infinitely dimensional spaces
A space is infinitely dimensional, if it has no basis consisting of finitely many vectors. By Zorn Lemma (see here), every space has a basis, so an infinite dimensional space has a basis consisting of infinite number of vectors (sometimes even uncountable).
It is worth noticing, that also in an infinite dimensional space the coordinates of a vector with respect to a given basis are unique, and a basis is a maximal linearly independent system of vectors, as well as minimal system of vectors spanning the considered space.
Intersection of subspaces
If and are subspaces of a space , then is a subspace of as well.
E.g. if , , then . If on the other hand and , then is the space of solutions to
Sum of subspaces
If and are subspace of , then is also a subspace of .
E.g. if , , then .
It is easy to notice that and that in the case of finite dimensional spaces .
We say that is a direct sum of its subspaces and (denoted by ), if , but also . In other words, every vector can be uniquely described as , where and . Obviously, if , then .
E.g. , but even though , this sum is not direct.