We have given the field $K$ and set $M$ and the $K$-vector space $V = \text{Abb}(M,K)$ (where $\text{Abb}(M,K)$ are all the maps from $M$ to $K$). Decide if the following sets $U$ are subspaces ($N\subset M$):
$$U = \{f\in \text{Abb}(M,K) : f(m) =0 ~\forall~ m \in M\setminus N\}$$
I tried to approach the problems with the subspace criteria, i.e. for all $f,g\in U$ and $\lambda \in K$ we have to have $f\circ g \in U$ and $\lambda \cdot f \in U$. However, I was not able to make any progress with this approach.
My approach for a: So I pick $f,g\in U$ and now I have to show that $f\circ g \in U$. But here I don't see how to continue. Also, I have to show that $\lambda \cdot f \in U$.