Isometrically isomorphic normed spaces

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In summary, if X and Y are normed spaces that are isometrically isomorphic, then their duals X' and Y' are also isometrically isomorphic. This means that for every element f in X', there exists a corresponding element G(f) in Y' such that the map G:X'->Y' is an isometric isomorphism. In order to find this map, we can use the property of duality that maps are contravariant, so instead of going from X' to Y', we can go from Y' to X' and still maintain an isometric isomorphism.
  • #1
iris_m
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Let X and Y be normed spaces. If X and Y are isometrically isomorphic, then their duals X' and Y' are also isometrically isomorphic.

I have no idea what to do with this, please help. :(
 
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  • #2
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
Duality if contravariant. Surely a morphism F:X->Y more naturally (no pun) gives a map G:Y'->X'.
 
  • #4
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'.
 

1. What is an isometrically isomorphic normed space?

An isometrically isomorphic normed space is a mathematical concept in which two normed spaces have the same structure and properties, except for possible differences in their names of the elements. Essentially, it is a way of comparing two spaces that have the same behavior, but may have different labels or symbols for the elements within them.

2. What is the difference between isometric and isomorphic?

Isometric and isomorphic are two related concepts, but with some key differences. Isometric refers to a mapping or transformation that preserves distances between points, while isomorphic refers to a mapping or transformation that preserves the structure and behavior of a space. In the context of normed spaces, isometric implies that the spaces have the same metric properties, while isomorphic implies that the spaces have the same algebraic and topological properties.

3. Can two normed spaces be isometrically isomorphic if they have different dimensions?

Yes, it is possible for two normed spaces to be isometrically isomorphic even if they have different dimensions. This is because the isomorphism is based on the structure and properties of the spaces, not the specific number or arrangement of elements within them. However, it is important to note that the dimensions of the spaces must be finite for them to be considered isometrically isomorphic.

4. How is isometrically isomorphic normed spaces related to linear transformations?

Isometrically isomorphic normed spaces are closely related to linear transformations, as they both involve preserving the structure and properties of a space. In fact, an isometric isomorphism can be thought of as a special type of linear transformation that preserves distances between points. In this sense, isometrically isomorphic normed spaces are a subset of linear transformations.

5. Are all isometrically isomorphic normed spaces equivalent?

No, not all isometrically isomorphic normed spaces are equivalent. While they may have the same structure and properties, they can still have differences in terms of specific elements or operations within the space. Additionally, even small changes in the definition of the norm or metric can result in non-equivalent isometrically isomorphic normed spaces. Therefore, it is important to carefully consider the specific properties and definitions of the spaces in question.

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