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.

Answer

  1. $\Bbb R[x]$, the polynomials in one variable.
  2. All the continuous functions from $\Bbb R$ to itself.
  3. 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.
  4. All the infinite sequences over $\Bbb R$.

And many many others.

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

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