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## Dual spaces for Quotients and Subspaces

Idea: to describe, for , and in terms of .

Definition 10.3.1

Remark:
If is a Hilbert space, then iff for all . That is, (Hilbert space definition).
So, this definition can be viewed a generalization of orthogonality for Hilbert spaces.

We can reformulate the 5th corollary of the Hahn-Banach theorem, if and only if for all .

Notice that so is a Banach space. We have defined by . Then,

We have:

with . Also, (using proposition at end of section on quotients). We also have:

given by where is the quotient map.

Theorem 10.3.2
1. gives an isometric isomorphism from onto
2. gives an isometric isomorphism from onto

Remark:

1. Dual of a subspace is a quotient of the dual space.
2. Dual of a quotient is a subspace of the dual space.

Proof. To see that is isometric, given by Hahn-Banach, there exists such that and . Then, , so is onto and .
Since , . Hence, . To see that , notice that and , so

maps to . So,
Given , . By the proposition, there is such that

So, .
To show is isometric, we need for all . Given , there is such that and Pick such that . Then,