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Adjoint

In mathematics, the term adjoint applies in several situations.

Adjoint

In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type

(Ax, y) = (x, By).

Specifically, adjoint or adjunction may mean:

  • Adjoint of a linear map, also called its transpose in case of matrices
  • Hermitian adjoint (adjoint of a linear operator) in functional analysis
  • Adjoint endomorphism of a Lie algebra
  • Adjoint representation of a Lie group
  • Adjoint functors in category theory
  • Adjunction (field theory)
  • Adjunction formula (algebraic geometry)
  • Adjunction space in topology
  • Conjugate transpose of a matrix in linear algebra
  • Adjugate matrix, related to its inverse
  • Adjoint equation
  • The upper and lower adjoints of a Galois connection in order theory
  • The adjoint of a differential operator with general polynomial coefficients
  • Kleisli adjunction
  • Monoidal adjunction
  • Quillen adjunction
  • Axiom of adjunction in set theory
  • Adjunction (rule of inference)

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What is a Hilbert space?

A vector space with an inner product (so lengths and angles make sense) that is complete (no missing limit points). Square-integrable functions form one; quantum states live in one.

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