I think these functions are seminorms but not norms (in "Undergraduate Analysis" by Serge Lang)

Question

I am reading "Undergraduate Analysis" by Serge Lang.
The above image is from "Problems and Solutions for Undergraduate Analysis" by Rami Shakarchi.

I think $\sigma_1 + \sigma_2$ and $\max(\sigma_1, \sigma_2)$ are seminorms but not norms.

Are both Serge Lang and Rami Shakarchi wrong?

Or am I wrong?

Answer 1

You are probably forgetting that $\sigma_1$ is assumed to be a norm. In both cases we have $\sigma_1(x)=0$ which implies that $x=0$.

Answer 2

The missing step is possibly going from $\sigma_1(x)+\sigma_2(x)=0$ to $\sigma_1(x)=\sigma_2(x)=0$.

Part of the definition of (semi)norm is that $\sigma(x)\ge0$, for every $x$. Now, if the sum of two nonnegative numbers is zero, then both are zero.

For the other one, from $\max\bigl(\sigma_1(x),\sigma_2(x)\bigr)=0$ it also follows $\sigma_1(x)=\sigma_2(x)=0$, because $$ 0\le\max\bigl(\sigma_1(x),\sigma_2(x)\bigr) $$ because of nonnegativity of (semi)norms.

Since $\sigma_1$ is assumed to be a norm, from $\sigma_1(x)=0$ we can deduce $x=0$.