 ##  [Inner Product](/inner-product) 

  ##  [Inner Product](https://quantummech.quantumdictionary.io/inner-product-1) 

  
 [![Quantum Mechanics Dictionary](/sites/default/files/styles/large/public/2026-01/Quantum%20Mechanics.png.webp?itok=tC_FGCW0)](/topic-specific-dictionaries/modern-physics/quantum-mechanics)

- 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](https://natural.quantumdictionary.io/inner-product-0) 

  
 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)


**Natural &amp; 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⟩ &gt; 0 for v ≠ 0).

 
 ##  [Inner Product](https://mathlogic.quantumdictionary.io/inner-product-2) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; 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().