Extension of equivalent norms (Exercise 2.4 in “Linear Analysis” by Bollobás) The following is Exercise 4 from Chapter 2 of Linear Analysis, an introductory course by Béla Bollobás.

Let $X = (V, \|\cdot\|)$ be a normed space and $W$ a subspace of $V$. Suppose $|\cdot|$ is a norm on $W$ which is equivalent to the restriction of $\|\cdot\|$ to $W$.
Show that there is a norm $\|\cdot\|_1$ on $V$ that is equivalent to $\|\cdot\|$ and whose restriction to $W$ is precisely $|\cdot|$.
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In a solution of this question, must the Hahn–Banach theorem be used used or can we prove it without using Hahn–Banach?
 A: Since the norms $\|\cdot\|$ and $|\cdot|$ are equivalent on $W$, for some $k$ we have $|w|\le k\|w\|$ for all $w\in W$. 
Let $B_1=\{x\in X:\|x\|\le1/k\}$ and $B_2=\{w\in W:|w|\le1\}$. Let $U$ be the convex hull of $B_1\cup B_2$. 

The plane of the diagram represents the subspace $W$. The central dot is the origin. The red pentagon is the boundary of the unit ball $B_2$. The green triangle is the boundary of the intersection of the unit ball $B$ wrt $\|\cdot\|$ and $W$. The smaller triangle is that dilated by $1/k$, ie the boundary of $B_1$. So $|w|\le1$ for any point on the smallest triangle and hence it lies entirely inside the red pentagon.
$B_2$ lies entirely in the plane, but there will be points of $B_1$ above and below the plane. So it is clear that $U\cap W=B_2$.
We take $U$ as the unit ball for the desired norm $\|\cdot\|_1$.
Since $U\cap W=B_2$, the norm $\|\cdot\|_1$ coincides with $|\cdot|$ on $W$.
Since we could also expand the green triangle to contain the red pentagon, it is clear that $\|\cdot\|$ and $\|\cdot\|_1$ are equivalent on $X$.
