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.

**Answer**

- $\Bbb R[x]$, 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.

**Attribution***Source : Link , Question Author : emDiaz , Answer Author : Asaf Karagila*