What are some examples of infinite dimensional vector spaces? I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.
 A: I think the following two examples are quite helpful:
For any field $F$,


*

*the set $F^{\mathbb N}$ of all sequences over $F$ and

*the set of all sequences over $F$ with finite support


are $F$-vector spaces.
Note that the unit vectors form a basis of the second vector space, but not of the first.
A: Think of function spaces, such as the continuous or differentiable or analytic functions on an interval.
A: *

*$\Bbb R[x]$, the polynomials in one variable.

*All the continuous functions from $\Bbb R$ to itself.

*All the differentiable functions from $\Bbb R$ to itself. Generally we can talk about other families of functions which are closed under addition and scalar multiplication.

*All the infinite sequences over $\Bbb R$.


And many many others.
A: I'm surprised to see that no one mentioned $\mathbb{R}$ over the field $\mathbb{Q}$ as a vector  space is infinite dimensional. Here is the proof.
A: *

*The space of continuous functions of compact support on a locally compact space, say $\mathbb{R}$.

*The space of compactly supported smooth functions on $\mathbb{R}^{n}$.

*The space of square summable complex sequences, commonly known as $l_{2}$. This is the prototype of all separable Hilbert spaces.

*The space of all bounded sequences.

*The set of all linear operators on an infinite dimensional vector space.

*The space $L^{p}(X)$ where $(X, \mu)$ is a measure space.

*The set of all Schwartz functions.


These spaces have considerable more structure than just a vector space, in particular they can all be given some norm (in third case an inner product too). They all fall under the umbrella of function spaces.
A: The two examples I like are these:
1) $\mathbb{R}[x]$, the set of polynomials in $x$ with real coefficients. This is infinite dimensional because $\{x^n:n\in\mathbb{N}\}$ is an independent set, and in fact a basis.
2) $\mathcal{C}(\mathbb{R})$, the set of continuous real-valued functions on $\mathbb{R}$. Here there is no obvious basis at all. This also has lots of interesting subspaces, some of which Hagen has mentioned.
