# Dual Space, Adjoint Operator

Definition (Dual Space)

The

dual space$X_{′}$ of a normed space $X$ is defined to be the vector space of all bounded linear functionals on $X$ together with the operator norm.

Definition (Adjoint Operator)

Suppose $X$ and $Y$ are normed spaces. If $T:X→Y$ is a bounded linear operator, then its

adjointis the operator $T_{′}:Y_{′}→X_{′}$ defined by $T_{′}f=f∘T$ for $f∈Y_{′}$.

We note some properties of the adjoint operator:

The adjoint operator $T_{′}$ is linear, bounded, and satisfies $∥T_{′}∥=∥T∥$.

If $S,T$ are bounded linear operators, and $α∈K$, then $(S+T)_{′}=S_{′}+T_{′}$ and $(αS)_{′}=αS_{′}$.

If $S:X→Y$ and $T:Y→Z$ are bounded linear operators, then $(TS)_{′}=S_{′}T_{′}$.

The proofs are straightforward.