# Proof of $\mathcal{L}(V,W)$ is a vector space

To prove whether something is a vector space, my understand is to prove the following properties: commutativity, associativity, additive identity, additive inverse, multiplicative identity, distributive properties. However, in the answer, it tries to prove closed under addition, close under scalar multiplication. are those conditions for a subspace?

• You are right that in the proof, they do not check commutativity, associativity etc. The picture you included shows that the vector space operations of addition and scalar multiplication $$L(V,W) \times L(V,W) \rightarrow L(V,W), (T,S) \mapsto T+S$$ $$\mathbb F \times L(V,W) \rightarrow L(V,W), (\lambda,T) \mapsto \lambda T$$ are well defined functions. This is the hardest part of showing that $L(V,W)$ is a vector space. You should check the other properties yourself. – D_S Jan 13 '19 at 22:20

Now, there is one potential saving grace here: perhaps it was proven that the set $$W^V$$ of functions from $$V$$ to $$W$$ (linear or not) is a vector space. That is, $$W^V$$ under the given operations, all of the 8 or so properties of vector spaces were proven previously. Then, proving $$\mathcal{L}(V, W)$$ is a subspace $$W^V$$ is a valid way of proving $$\mathcal{L}(V, W)$$ is a vector space in its own right, as it will inherit almost all the properties from $$W^V$$.