What does "the vector space of all polynomials on the interval of $[0,1]$" refer to?

Question

What might interval refer to in this context? I can't find a common quality between $x^2$ and $x+1$ that would make sense to qualify them as "being on the interval $[0,1]$". I've only ever heard of intervals in the context of intervals on a number line in $R^2$ for example. I read this in a linear algebra book so it's possible that it is related to linear algebra in some way.

Answer 1

Consider, for example, the set $V$ of differentiable functions, which map from $[0,1]$ to the real line. It's not hard to see that $V$ is a vector space with the zero function as its identity (intuition behind that being that linear combinations of differentiable functions are also differentiable).

Now, if you look at the $P \subset V$, which only considers the elements of $V$ which are polynomials, you will end up with the set your text is describing.

Answer 2

There is a distinction between polynomials, defined as linear combinations of the abstract objects $1,x,x^2,x^3,\cdots$, and polynomial functions, which are functions from some domain $D$ that happen to have a polynomial formula. There is a natural map from the former to the latter; we take a polynomial $f$ to the function $x\to f(x)$.
By specifying "on the interval $[0,1]$", we imply that we're talking about the latter space of polynomial functions.

Now, if that (one-variable, in a field) domain is infinite, the two spaces are isomorphic; the map from polynomials to polynomial functions is injective, since a polynomial with infinitely many zeros is the zero polynomial. This is the case for our example domain $[0,1]$.

Why would we specify that interval rather than the whole field $\mathbb{R}$? Most likely, we're going to go on to introduce some additional structure on the space related to that interval, such as the inner product $\langle f,g\rangle = \int_0^1 f(x)g(x)\,dx$ and the $2$-norm $\|f\|_2 = \sqrt{\int_0^1 f(x)^2\,dx}$. These wouldn't work on $\mathbb{R}$, since the integrals diverge for nonzero polynomials.