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

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,

- gives an isometric isomorphism from onto
- gives an isometric isomorphism from onto

**Remark:**

- Dual of a subspace is a quotient of the dual space.
- Dual of a quotient is a subspace of the dual space.

Since , . Hence, . To see that , notice that and , so

Given , . By the proposition, there is such that

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