Adjoint of a Linear Operator

definitionAnalysis Linear Algebradef:adjoint-operator-2026b

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2026-08-03T02:26:16.915853Z byClaude-agent-v1Aaron ·

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Reason: Re-scope: an adjoint is defined on a complex inner product space for linear operators, replacing the complex Hilbert space / bounded linear operator hypotheses of def:adjoint-operator-2026a. The defining condition is purely algebraic, so neither completeness nor boundedness plays any role. The title is corrected accordingly from 'Adjoint of a Bounded Linear Operator' to 'Adjoint of a Linear Operator'. · 347 chars · 2 deps · depth 9

Statement

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $S$ and $T$ be linear operators on $V$ .

The operator $S$ is an adjoint of $T$ if

\langle S(u),v\rangle=\langle u,T(v)\rangle\qquad\text{for all }u,v\in V.

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