# Definition:Adjoint Linear Transformation

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## Definition

Let $H, K$ be Hilbert spaces.

Let $A \in \map B {H, K}$ be a bounded linear transformation.

Let $B \in \map B {K, H}$ be the unique bounded linear transformation provided by Existence and Uniqueness of Adjoint.

Then $B$ is called the **adjoint** of $A$, and denoted $A^*$.

The operation of assigning $A^*$ to $A$ may be referred to as **adjoining**.

## Also see

- Existence and Uniqueness of Adjoint, which ensures this concept is well-defined.
- Definition:Self-Adjoint Operator
- Definition:Unitary Operator

- Results about
**adjoints**can be found here.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.2.4$