Question

Give an example of an infinite dimensional vector space

Answer 1

The set of all polynomials comes to mind offhand, as they can be of any order. Not sure if that counts officially though. JamesJ, Help!

Answer 2

Oh, the set of all real valued or continuous functions on some interval [a,b] should be one.

Answer 3

Yes, the polynomials will do it.

Another favorite example is the continuous functions on the real line,
\[C = C(\mathbb{R})\]

This is a vector space under addition defined by
(f+g)(x) = f(x) + g(x) (and thus f, g in C => f + g in C)
and scalar multiplication
(cf)(x) = c.f(x) (and thus f in C => cf in C for any real number c)
The zero vector is the zero function
0(x) = 0 for all x
It's trivial to see that f + 0 = f; that + is associate; and all the other vector space axioms.

Now, C does not have a finite basis. (Proof?)

Answer 4

Wouldn't know where to begin frankly.

Answer 5

Well, you'll learn it when you learn it. Not particularly important.