I try to understand why we call a vector space infinite dimensional vector space. As far as I know, the set of polynomials is an infinite dimensional vector space. I guess it is indeed, because it has infinitely many basis vectors: $$1,x,x^2,x^3,…$$ I hope this is what infinite dimensional vector space mean.
I wonder if there is any finite dimensional function space? For instance the set of functions, which functions have the form of $$ax+b$$ is a good example for finite dimensional function space? Here the basis vectors could be the identity function: $x$ and an arbitrary $C\in\mathbb R$ constant. Am I right? Sorry if its not the first question in the topic, I just want to understand it with my own examples.
Best Answer
A function space is a set $F$ of functions from a set $A$ into a set $B$. If the set $B$ is also a real vector space, then it makes sense to ask whether or not $F$ is a real vector space, with respect to the usual operations of sum and product by a scalar.
In particular, yes, the set of functions from $\mathbb R$ into $\mathbb R$ which are of the form $x\mapsto ax+b$ is a function space. Actually, it is a $2$-dimensional vectors space, by the reason that you mentioned (excpet that you should add that $C$ cannot be $0$).
And the set of all polynomial functions from $\mathbb R$ into $\mathbb R$ is an infinite-dimensional vector space, since it has an infinite basis: $\{1,x,x^2,\ldots\}$.