Every linear map between finite-dimensional inner product spaces has a unique adjoint

Depends on

• The adjoint $T^*:W\to V$ is characterised by $\langle Tv,w\rangle_W=\langle v,T^*w\rangle_V$
• Finite-dimensional Riesz representation: every functional is uniquely $v\mapsto\langle v,w\rangle$
• Inner products separate vectors, and the induced norm is homogeneous: $\lVert\lambda v\rVert=|\lambda|\lVert v\rVert$

Used by

• Certified basis-pursuit denoising by primal-dual hybrid gradient Algorithm
• Chambolle–Pock primal-dual hybrid gradient Algorithm
• Clean-ancilla zero-branch postselection Definition
• Exact recovery of an encoded quantum code Definition
• Finite adiabatic path model Definition
• Finite Kraus representation Definition
• Finite projective measurements Definition
• Finite quantum mutual, conditional, and conditional mutual information Definition
• Interior product on the exterior algebra Definition
• Projected-unitary encoding and singular vectors Definition
• Szegedy row-preparation map and discriminant Definition
• Dirac and rank-one coordinate check Example
• A finite instrument has an isometric coherent record Lemma
• A finite POVM defines a classical-output quantum channel Lemma
• A finite reversible transition operator is self-adjoint on weighted L2 Lemma
• A unitary error-basis change gives orthogonal error ranges Lemma
• Complete positivity implies Choi positivity Lemma
• Finite-dimensional linear isometries satisfy the adjoint identity Lemma
• Inverse compression for positive resolvents Lemma
• Isometric invariance of finite quantum relative entropy Lemma
• Kraus operators in an identity-channel representation are scalar Lemma
• Optimality relations for exact two-block ADMM subproblems Lemma
• Orthogonal error restrictions determine syndrome projections Lemma
• Preserving the logical marginal forces a gauge-only channel Lemma
• Primal and infeasibility certificates for basis pursuit denoising Lemma
• Relative modular representation of finite quantum relative entropy Lemma
• Szegedy reflections and walk operators are unitary Lemma
• Szegedy walk invariant subspaces from discriminant eigenvectors Lemma
• The Knill–Laflamme coefficient matrix is positive Lemma
• Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=λ̄ T^*, (ST)^*=T^*S^*, and T^**=T Proposition
• A Controlled Global Phase Becomes a Relative Control Phase Theorem
• A unit-vector outer product is a density operator Theorem
• An endomorphism is an orthogonal projection exactly when it is idempotent and self-adjoint Theorem
• An explicit syndrome recovery corrects a Knill–Laflamme error family Theorem
• An isometric quantum encoding gives an orthogonal code projector Theorem
• An isometry followed by partial trace is a channel Theorem
• Cosine-Sine Block Decomposition Theorem
• Dirac Adjoint Amplitude Translation Theorem
• Every finite-dimensional channel has a finite isometric dilation Theorem
• Exact recovery implies the Knill–Laflamme condition Theorem

…and 13 more results.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Sheldon Axler, Linear Algebra Done Right, 4th ed., discussion following definition 7.1 (standard reference, not scraped)
• Sergei Treil, Linear Algebra Done Wrong, §5.5.1 (standard reference, not scraped)