$Hom(V,W)$ remains unchanged when norms of $V$ and $W$ are replaced with equivalent norms. I was thinking about the following question from section 3.4 of Loomis and Sternberg's Advanced Calculus

The fact that $Hom(V,W)$ is unchanged when norms are replaced by equivalent norms can be viewed as a corollary of Theorem 3.3. Show that this is so.

Here we are taking $V$ and $W$ to be normed real vector spaces and $Hom(V,W)$ to be the space of bounded linear maps between them. The theorem referenced in the question is the following:
Given normed vector spaces $U,V,$ and $W$. Also, if $T \in Hom(U,V)$ and $S \in Hom(V,W)$ then $S \circ T \in Hom(U,W)$ and $\|S \circ T \| \leq \|S\| \|T \|$.
I'll denote a vector space $V$ with norm $p: V \rightarrow \mathbb{R}$ as the pair $(V, p)$. I understand that if we have that if  $p_{1}, p_{2}$ are equivalent norms on $V$ and $q_{1}, q_{2}$ are equivalent norms on $W$, then I believe that the question is asking us to show that $T \in Hom( (V,p_{1}), (W,q_{1}))$ implies that $T \in Hom((V, p_{2}), (W, q_{2}))$.
I'm not quite sure how to go about this.
 A: This answer just fills in a few details in Daniel Fischer's comment above. I felt maybe writing an answer would be helpful and good practice.
The book states that two norms $p,q: V \rightarrow \mathbb{R}$ are equivalent is there exists scalars $a, b \in \mathbb{R}^{>0}$  where for any $\xi \in V$, we have 


*

*$p(\xi) \leq aq(\xi)$

*$q(\xi) \leq bp(\xi)$


This is the same as saying there exists constants $c_{1}, c_{2} \in \mathbb{R}$ where $c_{1}q(\xi) \leq p(\xi) \leq c_{2}p(\xi)$. $\DeclareMathOperator{\id}{id}$
A linear map $T: V \rightarrow W$ is bounded when there exists a constant $c$, where for any vector $\xi \in V$, we have $\|T(\xi) \|_{W} \leq c\|\xi \|_{V}$. Now suppose that that we have two norms $p$ and $q$ on $V$ and the identity operator $\id:(V,p) \rightarrow (V,q)$ is bounded. This means that for any $\xi \in  V$, we have a $c \in \mathbb{R}^{>0}$ where $q(\id(\xi)) \leq c p(\xi)$. Also if the operator $\id:(V,q) \rightarrow (V,p)$ is bounded, there is a $c'$ where for any $\xi \in  V$, we have  $p(\id(\xi)) \leq c' q(\xi)$. This says exactly that the $p,q$ are equivalent norms. 
My question above was trying to show $T \in Hom((V,p_{1}), (W,q_{1})$ implies $T \in Hom((V, p_{2}), (W,q_{2})$ using the theorem above. We can view $T$ as an element of $Hom((V, p_{2}), (W,q_{2}))$ in the following manner:
$$(V,p_{2}) \xrightarrow{\id} (V,p_{1}) \xrightarrow{T} (W, q_{1}) \xrightarrow{\id} (W, q_{2})$$
The theorem then states that $T \in Hom((V, p_{2}), (W,q_{2}))$.
