I would like to understand what I must prove. Can you help me with this?
Question: Prove that, if $A, B$ are normed vector spaces then $(A\times B, \|\cdot \|)$ is a normed vector space.
Note: Note that $A, B$ are infinite-dimensional normed vector spaces.
My question: First, I must prove 8 properties of vector space, then how do I formalize, ie, how do I define what I want to prove? I want to prove that this vector space is only normalized with the sum, maximum and Euclidean norm. How do I prove that these norms are equivalent in a infinite-dimensional space like this? How do I define what I want to prove?
The norms mentioned are
- sum $\|(x,y)\|=\|x\|_A+\|y\|_B$;
- maximum $\|(x,y)\|= \max(\|x\|_A,\|y\|_B)$;
- euclidian $\|(x,y)\|=\sqrt{\|x\|_A^2+\|y\|_B^2}$;
Is there any way to prove the equivalence between this norm, than I can use just one to prove that $(A\times B, \|\cdot\|)$ is a normed space instead prove for these three norms?
I would like some help to understand the statement and correctly define everything I have to prove. Thank you in advance for any help.
I think the following should work. Let $\pi_A\colon A\times B\to A$ and $\pi_B\colon A\times B\to B$ the projections on the spaces $A$ respectively $B$. Observe, these are linear mappings.
Define a norm on $A\times B$ by $$ \|x\|_{A\times B} := \|\pi_A(x)\|_A+\|\pi_B(x)\|_B.$$
Then one can easily show, that it fulfills all three axioms of a norm. Of course, it is not the only one.
For example, we would like to show, that $\|(x,y) \|_\max:=\max(\|x\|_A,\|y\|_B)$ and $\|(x,y) \|_1:=\|x\|_A+\|y\|_B$ are equivalent.
If $\|x\|_A\geq \|y\|_B$, then we have $\|(x,y) \|_\max=\|x\|_A$ $$ \|(x,y) \|_\max=\|x \|_A\leq \|(x,y) \|_1=\|x\|_A+\|y\|_B\leq 2\|x\|_A=2\|(x,y) \|_\max. $$ Hence, $\|(x,y) \|_\max\leq \|(x,y) \|_1\leq 2\|(x,y) \|_\max$