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.

Let ${\displaystyle X}$ be a vector space over either the real numbers ${\displaystyle \mathbb {R} }$ or the complex numbers ${\displaystyle \mathbb {C} .}$ A real-valued function ${\displaystyle p:X\to \mathbb {R} }$ is called a seminorm if it satisfies the following two conditions:

These two conditions imply that ${\displaystyle p(0)=0}$ and that every seminorm ${\displaystyle p}$ also has the following property:

Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this is not necessary since it follows from the other two properties.

By definition, a norm on ${\displaystyle X}$ is a seminorm that also separates points, meaning that it has the following additional property:

A seminormed space is a pair ${\displaystyle (X,p)}$ consisting of a vector space ${\displaystyle X}$ and a seminorm ${\displaystyle p}$ on ${\displaystyle X.}$ If the seminorm ${\displaystyle p}$ is also a norm then the seminormed space ${\displaystyle (X,p)}$ is called a normed space.

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map ${\displaystyle p:X\to \mathbb {R} }$ is called a sublinear function if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is not necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem. A real-valued function ${\displaystyle p:X\to \mathbb {R} }$ is a seminorm if and only if it is a sublinear and balanced function.