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I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $R^n$ when thinking about vector spaces.

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The set of continuous functions $f:\mathbb{R} \mapsto \mathbb{R}$ is my go-to infinite dimensional vector space. –  FireGarden Aug 13 '13 at 16:15
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The $L^p$ spaces are infinite dimensional vector spaces. So are the bump functions and Schwartz functions. The vector space of entire functions is infinite dimensional as well. –  Cameron Williams Aug 13 '13 at 16:17
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5 Answers

up vote 6 down vote accepted
  1. The space of continuous functions of compact support on a locally compact space, say $\mathbb{R}$.
  2. The space of compactly supported smooth functions on $\mathbb{R}^{n}$.
  3. The space of square summable complex sequences, commonly known as $l_{2}$. This is the prototype of all separable Hilbert spaces.
  4. The space of all bounded sequences.
  5. The set of all linear operators on an infinite dimensional vector space.
  6. The space $L^{p}(X)$ where $(X, \mu)$ is a measure space.
  7. 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.

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  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.

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Concrete bunch of examples! –  B. S. Aug 13 '13 at 16:18
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Think of function spaces, such as the continuous or differentiable or analytic functions on an interval.

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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) $\mathcal{C}^\infty(\mathbb{R})$, 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.

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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.

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These are my favourites, too; the latter is cool because, while infinite, it tends to be quite well-behaved; whereas the former is cool because $\mathbb{N}$ is not a basis for $F^\mathbb{N}.$ –  user18921 Mar 23 at 9:53
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