seminorm
Sign in to saveIn mathematics, particularly in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
~17 min read
Article
19 sectionsContents
- Definition
- Examples
- Minkowski functionals and seminorms
- Algebraic properties
- Relationship to other norm-like concepts
- Inequalities involving seminorms
- Hahn–Banach theorem for seminorms
- Topologies of seminormed spaces
- Pseudometrics and the induced topology
- Stronger, weaker, and equivalent seminorms
- Normability and seminormability
- Topological properties
- Continuity of seminorms
- Continuity of linear maps
- Generalizations
- See also
- Notes
- References
- External links
In mathematics, particularly in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.
Connections
locally convex space
Entity
triangle inequality theorem
Entity
normed vector space
Entity
homogeneous function
Entity
Hahn–Banach theorem
Entity
linear functional
Entity
open mapping theorem
Entity
balanced set
Entity
convex cone
Entity
weak topology
Entity
reflexive space
Entity
mathematics
Entity
International Standard Book Number
Entity
complex number
Entity
real number
Entity
vector space
Entity
absolute value
Entity
OCLC, Inc.
Entity
metric space
Entity
Hilbert space
Entity