# Isometrically isomorphic normed spaces

1. Jun 21, 2008

### iris_m

Let X and Y be normed spaces. If X and Y are isometrically isomorphic, then their duals X' and Y' are also isometrically isomorphic.

2. Jun 21, 2008

### morphism

This is just a matter of writing down what you have, and understanding the definitions. Say you have an isometric isomorphism F:X->Y. We want to get an isometric isomorphism G:X'->Y'. The question you should be asking yourself is: If we start with an element f in X', how can we use this to get an element G(f) in Y'?

3. Jun 21, 2008

### matt grime

Duality if contravariant. Surely a morphism F:X->Y more naturally (no pun) gives a map G:Y'->X'.

4. Jun 21, 2008

### morphism

Right. I had initially written my G as going from Y' to X', but decided to stick with X'->Y' in case the OP doesn't immediately see why we've switched directions. But regardless, the moment he/she sets up the obvious map, he/she will probably notice that it's easier to go from Y' to X'.