##  [Norm](/norm) 

  ##  [Norm](https://natural.quantumdictionary.io/norm-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 & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A function ||·|| from a vector space V over R or C to the nonnegative reals satisfying (1) ||v|| ≥ 0 with ||v|| = 0 iff v = 0 (positive definiteness), (2) ||αv|| = |α|·||v|| for scalars α (absolute homogeneity), and (3) ||v+w|| ≤ ||v|| + ||w|| (triangle inequality).

 

 

 

 

 





 

 



 ##  [Norm](https://mathlogic.quantumdictionary.io/norm-1) 

  

 [![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 & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A real-valued function on a vector space that assigns a nonnegative size to each element and satisfies positive definiteness (zero only at the zero vector), absolute homogeneity (scaling by scalars), and the triangle inequality (subadditivity).

 

 

 

 

 





 

 



 ##  [Norm](https://puremath.quantumdictionary.io/norm-2) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A norm is a function ‖·‖ from a vector space to [0,∞) assigning each vector a nonnegative length and satisfying positive definiteness (‖v‖=0 iff v=0), absolute homogeneity (‖αv‖=|α|‖v‖), and the triangle inequality (‖u+v‖≤‖u‖+‖v‖).