I read a question about notation (you can find it here). One of the answers says that $V^{\mathbb{N}}$ where $V$ is vector space over a field $K$, is the space of functions $f: \mathbb{N} \longrightarrow V$. I'm still confused on how is this a vector space, also, I would like to know if such as $V^{\mathbb{N}}$ is a vector space, $V^{S(\mathbb{N})}$ is a vector space too. Going further, can $V^{\alpha}$ be a vector space, where $\alpha$ is an ordinal number.
P. S. I do apologize myself if there's another question that gives answer to mine (or if it's too trivial), I would appreciate if someone could post the link (and forgive me).
For $any$ non-empty $S,$ for any $f,g\in V^S,$ for any $s\in S,$ for any $k\in K,$ define $(f+'g)(s)=f(s)+g(s)$ and $(kf)(s)=k\cdot f(s).$ Then $V^S$ satisfies all the conditions in the definition of of a vector space over $K.$
For convenience the symbol $+$ is usually used for $+'$ as well as for addition of members of $K$.