I should give an example for an proper seminorm, so a norm for which every norm-axiom holds but $||x||=0 \Rightarrow x=0$. We assume that we use the Vectorspace $\mathbb{R}^n$ for our x. My guess would be $$ ||x||=|\sum_i^n x_i| $$ (N1) $||x+y||=|\sum_i^n (x_i+y_i)|=|\sum_i^n x_i+ \sum_i^n y_i|\leq |\sum_i^n x_i|+|\sum_i^n y_i|=||x||+||y||$
(N2)$||ax||=|\sum_i^n ax_i|=|a \sum_i^n x_i|=|a||\sum_i^n x_i|=|a| ||x||$
(N3/1)$||x||=|\sum_i^n x_i|\geq 0$
but for $x:=(1,-1,0,0,...0)^T\neq 0$ we have $||x||=|\sum_i^n x_i|=0$
Is that ok or did I do a mistake? I am unsure because in the answers of the book were a few examples but this one was not there. Thanks in advance :)
yes, your example is correct. You can also choose a more trivial example, like setting the norm of every vector to zero.