# Equivalence of two norms; Definition and Theorem from Kress

The following is from Linear Integral Equations by Kress.

1. Should definition $$1.4$$ read that "...if each sequence converging $${\color{red}{\text{to } x \in X}}$$ with respect to the first norm also converges $${\color{red}{\text{to the same }x}}$$ with respect to the second norm and vice versa."
2. In the Proof of Theorem $$1.5$$ for the second part, how to prove that if there is no $$C \in \mathbb{R}$$, then there exists a sequence $$(\phi_n)$$ with $$\Vert \phi_n\Vert_a = 1$$ and $$\Vert \phi_n \Vert_b \geq n^2$$? Is the following the right way to think about?

We will prove that if there doesn't exist $$c, C\in \mathbb{R}$$ such that $$c\Vert{\phi}\Vert_a \leq \Vert{\phi}\Vert_b \leq C\Vert{\phi}\Vert_a$$ for all $$\phi \in X$$, then the two norms $$\Vert{\cdot}\Vert_a$$ and $$\Vert{\cdot}\Vert_b$$ are not equivalent. Since there doesn't exist constants $$c$$ and $$C$$, this means there exists a sequence of $$\phi_n \in X$$, such that $$\dfrac{\Vert{\phi_n}\Vert_b}{\Vert{\phi_n}\Vert_a} \geq n^2$$. Consider the sequence $$\psi_n = \dfrac1n \cdot \dfrac{\phi_n}{\Vert{\phi_n}\Vert_a}$$. We then have $$\Vert{\psi_n}\Vert_a = \dfrac1n$$ and $$\Vert{\psi_n}\Vert_b \geq n$$. This means $$\lim_{n \to \infty} \Vert{\psi_n}\Vert_a = 0$$, whereas $$\lim_{n \to \infty} \Vert{\psi_n}\Vert_b \geq n$$. Hence, $$\psi_n \to 0$$ in $$\Vert{\cdot}\Vert_a$$, whereas $$\psi_n$$ doesn't converge to $$0$$ in $$\Vert{\cdot}\Vert_b$$.

1 The definition of equivalent I've seen taken is the existence of the $c$, $C$. i.e. two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ are equivalent if there are real constants $c, C>0$ such that for all $\phi$ in $X$ $$c\|\phi\|_a\leq\|\phi\|_b\leq C\|\phi\|_a$$
2 Yes your proof is correct. It's worth mentioning the trivial $X$ case in my opinion. Below is a slightly reformulated version of the same proof.
Suppose there is no $C>0$ such that for all $\phi\in X$ $$\|\phi\|_b\leq C\|\phi\|_a$$ Note that $X$ isn't trivial, since then any $C$ would satisfy $0\leq C\cdot 0$. In particular for each $n$ there is some $\psi_n$ such that $$\|\psi_n\|_b\not\leq n^2{\|\psi_n\|_a}.$$ Now defining $$\phi_n:=\frac{\psi_n}{\|\psi_n\|_a}$$ Then $$\|\phi_n\|_a=\frac{\|\psi_n\|_a}{\|\psi_n\|_a}=1$$ Using the absolute homogeneity property. And $$\|\phi_n\|_b=\frac{\|\psi_n\|_b}{\|\psi_n\|_a}\geq n^2.$$