Dual Space, Adjoint Operator

Definition (Dual Space)

The dual space of a normed space is defined to be the vector space of all bounded linear functionals on together with the operator norm.

Definition (Adjoint Operator)

Suppose and are normed spaces. If is a bounded linear operator, then its adjoint is the operator defined by for .

We note some properties of the adjoint operator:

The adjoint operator is linear, bounded, and satisfies .

If are bounded linear operators, and , then and .

If and are bounded linear operators, then .

The proofs are straightforward.