Is this the most general function vector space? One can prove that the space of $p$ integrable functions and that other types of spaces (sequence spaces etc) are vector spaces by proving that they are subspaces of a more general vector space. 
Define $\mathfrak{F} = \{ f: X \to Y \}$ where $Y$ is a vector space over a field $\mathbf K$ and $X$ is any set.  Then $\mathfrak{F}$ is a vector space over $\mathbf K$ with the pointwise addition and scalar multiplication defined as usual.
Is this the most general function vector space we can talk about? 
Thank you.
 A: If you want a function space $\mathcal F$ of functions $X\to Y$ to be a $\mathbb K$ vector space you need an addition and scalar multiplication to be defined in some way that makes sense. The only really correct way to do this is for each $x\in X$ have a pointwise defined addition and scalar multiplication defined on $\{f(x)\mid f\in \mathcal F\}$. In other words for $\mathcal F(x)$ to have a $\mathbb K$ vector space structure for each $x\in X$.
These vector spaces could be different spaces, for example you could have $X=\{0,1\}$ and $Y$ the disjoint union of $L^2$ and $L^\infty$ and your function space consists of functions that map $0$ to a vector in $L^2$ and $1$ to a vector in $L^\infty$.
But this is not usually very sensible, and you could consider $Y$ as a subspace of $\bigoplus_{x\in X}\mathcal F(x)$ to get $\mathcal F$ to be a subspace of the form you have considered, but then you have to identify all the different zeros, which may make the functions mean something different.
As an aside, every vector space is a function space:
Let $V$ be a $\mathbb K$ vector space where $\mathbb K$ is some field. Every vector space has a Hamel-basis, let $\{e_i\}_{i\in I}$ be a Hamel-basis of $V$. Consider the vector space $\mathcal F(I)$ of functions $I\to\mathbb K$ with finite support. The map
$$\mathcal F(I) \to V, \qquad f \mapsto \sum_{i\in I} f(i) e_i$$
is an isomorphism of vector spaces.
