Why isometric isomorphic between Banach spaces means we can identify them?

Is the "isometric" part really necessary? For what reason is that?

Eg. we prove that there is an isometric isomorphism between $(L^p)'$ and $L^q$ ($(p,q)$ conjugate) and then we identify them together as the same space. If they were isomorphic but not necessarily isometric, could we not identify them?

Solution 1:

Each mathematical theory study their own objects and maps between them. In our case these are Banach spaces and bounded linear maps. We choose linear maps because we want them to preserve linear structure of Banach spaces. We choose them bounded to nicely interact with the norm structure of Banach spaces. Once we choose our objects and transformation between them we want to classify our objects. There mathematicians start to invent different properties which can be used for classification. The more properties two Banach spaces have in common the more they are similar.

Now we want to know what are the maps that preserve a particular property. The best maps in Bаnach space theory are

• isomorphisms, i.e. bijective bounded linear operators with bounded inverse

• isometric isomorphisms, i.e. isomorphisms that preserve norms

The first type of isomorphisms preserve the following properties

• complementability

• separablity

• completeness

• type/cotype

• reflexivity

and many other that depend on the topology generated by the norm. Since topology of Banach spaces can be generated by different norms isomorphisms of Banach spaces can not catch all properties of the norm structure of Banach space.

In this case the second type of isomorphisms will be useful. They preserve all properties mentioned above plus

• 1-complementabilty

• extreme points

• distances between susbsets

and others that depend on the metric generated by norm.

These types of isomorphisms give two ways of identification of Banach spaces - tolerant and more rigorous. For example $\ell_1^n$ and $\ell_2^n$ are two $n$-dimensional Banach spaces that isomorphic but not isometrically isomorphic.

Note that isomorphisms of Banach spaces catch only those properties of spaces that correspond to their Banach space structure. For example the spaces of convergent sequences $c$ is isomorphic as Banach space to the space $c_0$ of sequences converging to $0$. But as Banach algebras (Banach spaces with continuous multiplication of vectors) they are not isomorphic, because $c$ is unital and $c_0$ is not.

Solution 2:

We can identify two objects if there is a canonical isomorphism between them. The notion of isomorphism depends on context. In the theory of Banach spaces there are two common notions of an isomorphism:

• Bijective linear map $T:X\to Y$ such that both $T$ and $T^{-1}$ are bounded operators.
• Bijective linear map $T:X\to Y$ which is an isometry: $\|Tx\|=\|x\|$ for all $x$.

The second kind of isomorphism, called isometric isomorphism, preserves all the structure that a Banach space has. Hence, we lose nothing in identifying two spaces that are isometrically isomorphic to each other, provided that the isomorphism is canonical (e.g., does not depend on a choice of basis).

The first kind, called simply an isomorphism, throws out a bit of information. For example, one of two isomorphic spaces may be uniformly convex while the other one isn't. Generally, it is not practical to think of two isomorphic Banach spaces as being the same.