Suppose a vector space $V$ has infinite dimension, $\dim V = \infty$. Is the cardinality of the basis of $V$ neccesarily countable infinite?
If a vector space has infinite dimension does $V$ then have a basis ? If one can specify a set of infinitely many linearly independent vectors is this set then regarded as a basis for $V$ ?
Are there results that can be proven for vector spaces with finite dimension, but not for vector spaces with infinite dimension ?
Why does most books on linear algebra only mention finite dimension vector spaces ?
Best Answer
Not all infinite dimensional vector spaces have countable basis. In fact, most spaces you come across in "real life" don't. The simplest example would be a sequence space like $$ \ell^2 = \{(x_n) \subset \mathbb{C} : \sum_{n=1}^{\infty} |x_n|^2 < \infty\} $$ This is an example of a Banach space - a vector space that comes with a nice notion of distance (a norm) and one that is complete with respect to that norm.
It is a general fact that an infinite dimensional Banach space cannot have a countably infinite basis (due to the Baire Category theorem).
However, the space $$ c_{00} := \{(x_n) \in \mathbb{C} : x_n \neq 0 \text{ for only finitely many } n\} $$ is an example of an infinite dimensional vector space whose basis is countable (the "standard" basis).
Most books on Linear Algebra mention only finite dimensional vector spaces because they are easy to visualize (just extend your notion of a vector in $\mathbb{R}^2$), but they are also deep enough to prove some rather interesting results (for instance the spectral theorem). Furthermore, infinite dimensional vector spaces are best analysed with some topology in mind - this is what functional analysis studies.