I'm reading Luenberger's Optimization by Vector Space Methods which has the following definition:
A finite set $S$ of linearly independent vectors is said to be a basis for the space $X$ if $S$ generates $X$. A vector space having a finite basis is said to be finite dimensional. All other vector spaces are said to be infinite dimensional.
Am I going crazy for thinking that this definition does not really allow for infinite dimensional spaces since $S$ is defined as a finite set? Maybe there's some subtlety I'm missing here?
Best Answer
That definition makes sense. Take, for instance, the vector space $\Bbb R^{\Bbb N}$ of all sequences of real numbers. There is no finite set $S\subset\Bbb R^{\Bbb N}$ which spans $\Bbb R^{\Bbb N}$. Therefore, $\Bbb R^{\Bbb N}$ is infinite-dimensional.
There is a problem with that definition however: it doesn't allow the existence of bases of infinite-dimensional vector spaces.