Definitions. Let $X$ be any non-empty set and $\mathbb{F}$ a field. We define:
1) For any $f:X\to\mathbb{F}, \text{ supp}(f) := \{x\in X:f(x)\ne0\}$
2) $\mathcal{F}_X:=\{f:X\to \mathbb{F}, \text{ |supp}(f)|<\infty\}$
From the definitions above I have the following facts:
1) $\mathcal{F}_X$ is a vector space over $\mathbb{F}$ along with usual function sum and scalar product.
2) The function $\theta:X\to\mathcal{F}_X$ given by $\theta(x)=f_x$, where $f_x(y) = \delta_{xy}$ is injective.
3) $S = (f_x)_{x\in X}$ is a basis for $\mathcal{F}_X$
Now I have a theorem for universal property for vector spaces. It's from my class notes, but I think it is not correctly stated and/or complete, especially the (b) part.
Theorem (Universal Property). Let $\mathbb{F}$ be a field and $X$ any non-empty set. $k_0$ the inclusion from $X$ into $\mathcal{F}_X$. Then
a) If $W$ is a vector space and $t:X\to W$, then there exists a unique $\tau_0\in \mathcal{L}(\mathcal{F}_X,W)$ such that $\tau_0\circ k_0 = t$ (for this we say that the diagram with the spaces $X, \mathcal{F}_X$ and $W$ commutes). This remains true if we replace $\mathcal{F}_X$ with a vector space $U$ and $X$ being a base of $U$.
b) Let $V$ be a vector space. If for all $t: X\to W$ and $W$ vector space, the following diagram commutes, then $V$ is isomorphic to $\mathcal{F}_X$.
I stated exactly as it is in my notes. I'm not sure what the $k$ function is, but I think it's any map from the "basis" $X$ of $\mathcal{F}_X$ into the space $W$ and from the isomorphism and the fact that $k_0$ is injective we can conclude that $k$ is injective (so $k(X)$ would be a basis for $V$). Am I right? There must be given additional hypothesis or there's something to be corrected in the theorem?
Thanks in advance. (P.s. sorry for the poor diagram, I was not able to build one using MathJax =< )
Let $V$ be a vector space and $k:X\rightarrow V$, if for every map $t:X\rightarrow W$ there exists a unique linear map $\tau:V\rightarrow W$ such that $t=\tau\circ k$ then $V$ is isomorphic to $F_X$. In fact the free vector space generated by $X$ is the initial object of the category whose objects are $f:X\rightarrow U$ where $U$ is a vector space, a morphism between $f:X\rightarrow U$ and $g:X\rightarrow V$ is a linear map $h:U\rightarrow V$ such that $h\circ f=g$. Its for this reason that it is the couple $(V,k:X\rightarrow V)$ which is considered here.