Dual Space, Adjoint Operator

Definition (Dual Space)

The dual space $X^{\prime }$ 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 $T:X\to Y$ is a bounded linear operator, then its adjoint is the operator $T^{\prime }:Y^{\prime }\to X^{\prime }$ defined by $T^{\prime }f=f\circ T$ for $f\in Y^{\prime }$.

We note some properties of the adjoint operator:

The adjoint operator $T^{\prime }$ is linear, bounded, and satisfies $\Vert T^{\prime }\Vert =\Vert T\Vert$.

If $S,T$ are bounded linear operators, and $\alpha \in \mathbb{K}$, then $(S+T{)}^{\prime }=S^{\prime }+T^{\prime }$ and $(\alpha S{)}^{\prime }=\alpha S^{\prime }$.

If $S:X\to Y$ and $T:Y\to Z$ are bounded linear operators, then $(TS{)}^{\prime }=S^{\prime }{T}^{\prime }$.

The proofs are straightforward.