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.
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$\begingroup$- $\Bbb R[x]$, 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.
$\endgroup$ 0 $\begingroup$- 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.
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ 0 $\begingroup$Think of function spaces, such as the continuous or differentiable or analytic functions on an interval.
$\endgroup$ $\begingroup$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.
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