Serlo: EN: Function spaces

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In this article we consider the space of functions, that is, the vector space of all maps $f:X\to V$ of a set $X$ into a vector space $V$.

Definition of function spaces

Let $K$ be a field, $(V,{+}_V,{\cdot }_V)$ a $K$-vector space and $X$ some set.

Then we can define the set of maps of $X$ to $V$:

On this set we define an addition and a scalar multiplication:

The set of differentiable functions $f:(0,1)\to ℝ$ an an $ℝ$-vector space

In the previous section we showed that the set of all maps of a set $X$ into a $K$-vector space $V$ is again a $K$-vector space. We now consider the special case $X=(0,1)\subseteq ℝ$, $K=ℝ$ and $V=ℝ$. We already know that $V$ is a $K$-vector space. Hence, we know so far that the set of maps $f:(0,1)\to ℝ$ is an $ℝ$-vector space.

We now consider the set of differentiable functions $f:(0,1)\to ℝ$, which is denoted $𝒟((0,1),ℝ)$ (as "differentiable").

Mathe für Nicht-Freaks: Vorlage:Satz

Relation to the sequence space

We have already seen that the set of sequences over $K$ forms a vector space with respect to coordinate-wise operations. So a sequence $(a_n{)}_n$ with entries in $K$ can be seen as a function $\mathbb{N}\to K,n\mapsto a_n$. In this sense, the sequence space is a special case of the function space $\mathrm{Fun}(X,V)$ by setting $X:=\mathbb{N}$ and $V:=K$.

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