# Definition:Adjoint (Norm Theory)

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## Definition

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

Let $A: X \to Y$ be a linear operator between $X$ and $Y$.

Let $A^*: Y^* \to X^*$ be the linear operator defined as:

- $\forall x, y \in Y: \innerprod {A x} y = \innerprod x {A^* y}$

Then $A^*$ is the **adjoint** of $A$.

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## Also known as

$A^*$ is also known as the **dual** of $A$.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**adjoint**:**1. b.**