Real Analysis, Folland Problem 5.3.32 The Baire Category Theorem

Question

Problem 5.3.32 - Let $\lVert x \rVert_{1}$ and $\lVert x \rVert_{2}$ be norms on the vector space $\mathscr{X}$ such that $\lVert x \rVert_{1}\leq \lVert x \rVert_{2}$. If $\mathscr{X}$ is complete with respect to both norms, then the norms are equivalent.

I know that $0\in \mathscr{X}$ and also I know that the norms are equivalent if there exists a $C_1,C_2 > 0$ such that $$C_1\lVert x \rVert_{1} \leq \lVert x \rVert_{2} \leq C_2\lVert x \rVert_{1}$$ but I am not sure how to show this, any suggestions is greatly appreciated.

Answer 1

As you have said, you have to prove that there exist $C_1,C_2 > 0$ such that $$C_1\lVert x \rVert_{1} \leq \lVert x \rVert_{2} \leq C_2\lVert x \rVert_{1}$$

By hypothesis, we have $$C_1\lVert x \rVert_{1} \leq \lVert x \rVert_{2}\tag{1},$$ where $C_1=1$. So, it's enough to prove that there exists $C_2 > 0$ such that $$\lVert x \rVert_{2} \leq C_2\lVert x \rVert_{1}$$

In other words, you have to prove that the mapping $F:(\mathscr{X},\|\cdot\|_1)\to (\mathscr{X},\|\cdot\|_2)$ given by $F(x)=x$ is continuous.

Note that the mapping $F$ is the inverse of the mapping $T:(\mathscr{X},\|\cdot\|_2)\to (\mathscr{X},\|\cdot\|_1)$ given by $T(x)=x$.

Because of $(1)$, $T$ is continuous. So, by Corollary 5.11 of Folland, $F$ is continuous as desired.