Inner Product
- Modern Physics -
Quantum Mechanics Dictionary
Definition
A mathematical structure used to represent quantum states and the operations performed on them. It specifies how states are expressed, related, combined, and decomposed into components that support calculation. It does not by itself determine physical predictions without a mapping to observables and measurement rules.
Inner Product
Natural & Formal Sciences Dictionary
Definition
A bilinear (over R) or sesquilinear (over C) form ⟨·,·⟩: V×V → F on a vector space V over field F (R or C) that is symmetric (⟨v,w⟩ = ⟨w,v⟩) or Hermitian (⟨v,w⟩ = conjugate(⟨w,v⟩)) and positive definite (⟨v,v⟩ > 0 for v ≠ 0).
Inner Product
- Natural & Formal Sciences -
Mathematics & Logic Dictionary
Definition
A bilinear (over R) or sesquilinear (over C) positive-definite form on a vector space that pairs two vectors to produce a scalar, satisfying conjugate symmetry, linearity in one slot, and positivity, and which induces a norm via ||v|| = sqrt().