User:Caliburn/s/fa/Definition:Adjoint Linear Transformation/Normed Vector Space

Definition

Let $X$ and $Y$ be normed vector spaces.

Let $X^\ast$ and $Y^\ast$ be the normed dual spaces of $X$ and $Y$ respectively.

Let $T : X \to Y$ be a bounded linear transformation.

We define the adjoint linear transformation $T^\ast : Y^\ast \to X^\ast$ by:

$\map {\paren {T^\ast g} } x = \map g {T x}$

for each $g \in Y^\ast$ and $x \in X$.

In bracket notation, this condition can be expressed as:

$\innerprod {T x} g = \innerprod x {T^\ast g}$

for each $x \in X$ and $g \in Y^\ast$.

Also see

• Adjoint of Bounded Linear Transformation between Normed Vector Spaces is Well-Defined
• Adjoint of Bounded Linear Transformation between Normed Vector Spaces is Bounded Linear Transformation
• Norm of Adjoint: Normed Vector Space

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $4.10$: Theorem