Question

1 (20 pts) Let (V 1) and (w 1 D be normed spaces. Show that the following the following ll lli and 11 IL : V × W → R given by b) ll(t, te)l1 -max(11에, llull l (v.tr)EV ×W) are norms on V × W.

Answer 1

(a)

$||(v,w)||_1=||v||+||w||,(v,w)\epsilon V\times W$ is norm :

(1)

$Take\\c\epsilon \mathbb{R}\\ ||c(v,w)||_1=||(cv,cw)||=||cv||+||cw||=|c|||v||+|c|||w||=|c|||(v,w)||$

(2)

$||(v,w)+(v',w')||\\=||(v+v',w+w')||_1\\=||w+w'||+||v+v'||\leq ||w||+||w'||+||v||+||v'||\because ||||-norm\\$

                                                      $=||u||+||v||+||u'||+||v'||\\=||(u,v)||_1+||(u',v')||_1$

that is

$||(v,w)+(v',w')||_1\leq ||(v,w)||_1+||(v',w')||_1$

(3)

$||(v,w)||_1= ||v||+||w||\geq 0\because ||v||,||w||\geq 0$

(4)

If

$||(v,w)||_1=0\\ ||v||+||w||=0\\ \Rightarrow ||v||=0\&||w||=0\\||v||=0\Rightarrow v=0\\ \&\\||w||=0\Rightarrow w=0\\ (v,w)=0$

hence norm 1 is norm .

(b)

$||(u,v)||_\infty =max\left \{ ||v||,||w||:(v,w)\epsilon V\times W \right \}$

(1)

Take c in R,

$||c(v,w)||_\infty \\=||(cv,cw)||_\infty \\=max\left \{ ||cv||,||cw||:(v,w)\epsilon V\times W \right \}\\=max\left \{ c||v||,c||w|| \right \}\\=c*max\left \{ ||v||,||w|| \right \}\\=c||(v,w)||_\infty$

(2)

$||(v,w)+(v',w')||_\infty \\=||(v+v',w+w')||_\infty \\=max\left \{ ||v+w||,||v'+w'|| \right \}\\\leq max\left \{ ||v||,||w|| \right \}+max\left \{ ||v'||,||w'|| \right \}$.

(3)

$||(v,w)||_\infty \\=max\left \{ ||v||,||w|| \right \}\geq 0(\because ||v||,||w||\geq 0)$

(4)

If

$||(v,w)||_\infty \\=0\\max\left \{ ||v||,||w|| \right \}=0\Rightarrow \\||v||=0,|\w||=0\Rightarrow v=0,w=0\Rightarrow \\(v,w)=0$

hence infinite norm is norm.