To understand the distance function in (X, || . || ) × ( X, ||. || )

Question

A very silly doubt but I don't know what I am thinking. See , book introduced the the notion of inner product and then also defined how we can get norm from it. Then book is saying <.,.> : (X, || . || ) × ( X, ||. || ) $\to$ F(scalar field) is continuous. I don't want proof of this statement. What confusing me is how we have defined distance between two points in (X, || . || ) × ( X, ||. || ). Means say we have $(x_{1},x_{2})$ and $(x_{3},x_{4})$ from (X, || . || ) × ( X, ||. || ), then what is the distance between them i.e. what I want is what is the distance function in vector space (X, || . || ) × ( X, ||. || ).

Answer 1

Some examples of norm on $X \times X$ are $\|(x,y)\|=\|x\|+\|y\|$, $\|(x,y)\|=(\|x\|^{2}+\|y\|^{2})^{1/2}$, $\|(x,y)\|=\max (\|x\|,\|y\|)$. They all give rise to the same topology on $X \times X$.

Answer 2

Most common norms on the product space are probably those two (equivalent) norms: $$\lVert (x_1,x_2) \rVert_\infty=\max (\lVert x_1 \rVert, \lVert x_2 \rVert)$$ $$\lVert (x_1,x_2)\rVert_1= \lVert x_1 \rVert + \lVert x_2 \rVert$$ Distance between two points is as usual just the norm of the difference.