Explanation of a problem about vector space

Please could you explain for me the following problem. For me it is all about the understanding of the notation not solving the problem:

for f,g: M → K we define

f+g: M → K by 􏰁f+g􏰂(x) = f(x)+g(x) and for k ∈ K we define

k·f : M → K by 􏰁k·f􏰂(x) = k·f(x)

Prove that abb(M, K) := {f : M → K mapping} has a structure of a K-vector space.

The space Abb$(M,K)$ is a set with mappings as elements. So if we want to define a vector-space structure on this set, the mappings are now called "vectors".

If you want to equip the space Abb$(M,K)$ with a $K$-vector-space structure you need to define how to add two elements of this space - so you have to define an addition of two mappings $f,g: M \to K$.

The definition you gave is the pointwise definition of addition. It means that the element $f \oplus g$ in the space Abb$(M,K)$ is defined by how it works as a mapping, namely: The mapping $f \oplus g$ maps a point $x \in M$ to the sum of the points $f(x)$ and $g(x)$ in $K$.

It is similar for the scalar multiplication. The definition given here is again pointwise, so the element $k \otimes f$ for a mapping $f \in $ Abb$(M,K)$ and a $k \in K$ is defined by how it operates as a mapping. The mapping $k \otimes m$ maps a point $x \in M$ to the product of $k$ and $f(x)$ in $K$.