I am reading through Jim Hefferon's ["Linear Algebra"](http://joshua.smcvt.edu/linearalgebra/) and have encountered a lemma which I have had some difficulty in understanding. Through reading various other resources, I believe I currently understand the lemma but I am not complete sure. I would appreciate it if someone could confirm or correct my understanding.

The lemma is located near the end of Chapter 3, Section II.1 and reads as follows:

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> For vector spaces V and W, the set of linear functions from V to W is itself a vector space, a subspace of the space of all functions from V to W.
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> 
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What I believe this means is that there is a set of linear functions, $G = \{G\_1, G\_2, ...G\_n\}$ that map $V$ to $W$ such that these maps conform to the rules of vector spaces. As an example for all $G\_k, G\_m \in G,$ we would have closure: $G\_k+G\_m \in G,$ associativity under addition: $G\_k+G\_m = G\_m+G\_k$, additive identity: $G\_k + G\_n = G\_k, G\_n \in G$, etc.

Further, I believe the portion of lemma that reads:

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> "... subspace of the space of all functions from V to W."
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states that there are some non-linear functions that map from $V$ to $W$ and that the functions we would include in $G$ (as defined in the previous paragraph) would be the subset of of these functions which are linear.

Is this a correct interpretation?