Dual Norm Spaces (isomorphism/isometric)

[CornMuffin]

Homework Statement

If X and Y are normed spaces, define $\alpha : X^* x X^*\rightarrow (X x X)^*$ by $\alpha(f,g)(x,y) = f(x)+g(y)$.
Then $\alpha$ is an isometric isomorphism if we use the norm $||(x,y)|| = max(||x||,||y||)$ on $X x Y$, the corresponding operator norm on $(X x Y)^*$, and the norm $||(f,g)||=||f||+||g||$ on $X^* x Y^*$.

Homework Equations

$||x||=sup[|f(x)|:f\in X^*, ||f||\leq 1]$
$||f||=sup[|f(x)|:x\in X, ||x||\leq 1]$

The Attempt at a Solution

to show it is isomorphism,
suppose $\alpha (f,g) = f(x) + g(y) = m(x) + n(y) = \alpha (m,n)$
but this can only happen if f(x)=m(x) and g(y)=n(y) since they depend on x and y respectfully.

but i am having trouble with proving it is isometric
here is my attempt:
I assume i want to show that ||(f,g)||=||(x,y)||

$||(f,g)||=sup[|f(x)|:x\in X, ||x||\leq 1] + sup[|g(y)|:y\in Y, ||y||\leq 1] =sup[|f(x)|+|g(y)|:x\in X, y\in Y, ||y||\leq 1, ||x||\leq 1]$


also
$||(x,y)||=max(sup[|f(x)|:f\in X^*, ||f||\leq 1],sup[|g(y)|:g\in X^*, ||f||\leq 1])$

but I don't know what else to do.

 

[mighty2000]

First, we need to show that \alpha is a linear map. Let (f1,g1), (f2,g2) be elements of X^* x Y^*. Then for any (x,y)\in X x Y, we have:

\alpha((f1,g1)+(f2,g2))(x,y) = (f1+f2)(x)+(g1+g2)(y) = f1(x)+f2(x)+g1(y)+g2(y)
= (f1(x)+g1(y))+(f2(x)+g2(y)) = \alpha(f1,g1)(x,y)+\alpha(f2,g2)(x,y)

So, \alpha is indeed a linear map. Now, let's show that it is an isometry. For any (f,g)\in X^* x Y^*, we have:

||(f,g)|| = ||f||+||g|| = sup[|f(x)|:x\in X, ||x||\leq 1] + sup[|g(y)|:y\in Y, ||y||\leq 1]
= sup[|f(x)|+|g(y)|:x\in X, y\in Y, ||y||\leq 1, ||x||\leq 1]

On the other hand, for any (x,y)\in X x Y, we have:

||(x,y)|| = max(||x||,||y||) = sup[|f(x)|:f\in X^*, ||f||\leq 1] + sup[|g(y)|:g\in X^*, ||g||\leq 1]
= sup[|f(x)|+|g(y)|:f,g\in X^*, ||f||\leq 1, ||g||\leq 1]

Since both ||(f,g)|| and ||(x,y)|| are equal to sup[|f(x)|+|g(y)|:x\in X, y\in Y, ||y||\leq 1, ||x||\leq 1], we can conclude that \alpha is an isometry. This means that it preserves distances between elements in X^* x Y^*, and hence is an isometric isomorphism.