seminormed space

In mathematics, particularly in functional analysis, a seminorm is a vector space norm that 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.

Definition

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

# Subadditivity/ Triangle inequality: $p(x + y) \leq p(x) + p(y)$ for all $x, y \in X.$
# Absolute homogeneity: $p(s x) =, s, p(x)$ for all $x \in X$ and all scalars $s.$

These two conditions imply that $p(0) = 0$If $z \in X$ denotes the zero vector in $X$ while $0$ denote the zero scalar, then absolute homogeneity implies that $p(z) = p(0 z) = , 0, p(z) = 0 p(z) = 0.$ $\blacksquare$ and that every seminorm $p$ also has the following property:Suppose $p : X \to \R$ is a seminorm and let $x \in X.$ Then absolute homogeneity implies $p(-x) = p((-1) x) =, -1, p(x) = p(x).$ The triangle inequality now implies $p(0) = p(x + (- x)) \leq p(x) + p(-x) = p(x) + p(x) = 2 p(x).$ Because $x$ was an arbitrary vector in $X,$ it follows that $p(0) \leq 2 p(0),$ which implies that $0 \leq p(0)$ (by subtracting $p(0)$ from both sides). Thus $0 \leq p(0) \leq 2 p(x)$ which implies $0 \leq p(x)$ (by multiplying thru by $1/2$).

3. Nonnegativity: $p(x) \geq 0$ for all $x \in X.$

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 $X$ is a seminorm that also separates points, meaning that it has the following additional property:

4. Positive definite/: for all $x \in X,$ if $p(x) = 0$ then $x = 0.$

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

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map $p : X \to \R$ is called a if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem.
A real-valued function $p : X \to \R$ is a seminorm if and only if it is a sublinear and balanced function.

Examples

• The on $X,$ which refers to the constant $0$ map on $X,$ induces the indiscrete topology on $X.$


• If $f$ is any linear form on a vector space then its absolute value $, f, ,$ defined by $x \mapsto , f(x), ,$ is a seminorm.


• A sublinear function $f : X \to \R$ on a real vector space $X$ is a seminorm if and only if it is a , meaning that $f(-x) = f(x)$ for all $x \in X.$


• Every real-valued sublinear function $f : X \to \R$ on a real vector space $X$ induces a seminorm $p : X \to \R$ defined by $p(x) := \max \.$


• Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a vector subspace is once again a seminorm (respectively, norm).


• If $p : X \to \R$ and $q : Y \to \R$ are seminorms (respectively, norms) on $X$ and $Y$ then the map $r : X \times Y \to \R$ defined by $r(x, y) = p(x) + q(y)$ is a seminorm (respectively, a norm) on $X \times Y.$ In particular, the maps on $X \times Y$ defined by $(x, y) \mapsto p(x)$ and $(x, y) \mapsto q(y)$ are both seminorms on $X \times Y.$


• If $p$ and $q$ are seminorms on $X$ then so are
  $(p \vee q)(x) = \max \ \quad \text \quad (p \wedge q)(x) := \inf \$
  where $p \wedge q \leq p$ and $p \wedge q \leq q.$
  


• The space of seminorms on $X$ is generally not a distributive lattice with respect to the above operations. For example, over $\R^2$, $p(x, y) := \max(, x, , , y, ), q(x, y) := 2, x, , r(x, y) := 2, y,$ are such that$((p \vee q) \wedge (p \vee r)) (x, y) = \inf \ \quad \text \quad (p \vee q \wedge r) (x, y) := \max(, x, , , y, )$


• If $L : X \to Y$ is a linear map and $q : Y \to \R$ is a seminorm on $Y,$ then $q \circ L : X \to \R$ is a seminorm on $X.$ The seminorm $q \circ L$ will be a norm on $X$ if and only if $L$ is injective and the restriction $q\big\vert_$ is a norm on $L(X).$

Minkowski functionals and seminorms

Seminorms on a vector space $X$ are intimately tied, via Minkowski functionals, to subsets of $X$ that are convex, balanced, and absorbing. Given such a subset $D$ of $X,$ the Minkowski functional of $D$ is a seminorm. Conversely, given a seminorm $p$ on $X,$ the sets$\$ and $\$ are convex, balanced, and absorbing and furthermore, the Minkowski functional of these two sets (as well as of any set lying "in between them") is $p.$

Algebraic properties

Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including:
* Convexity
* Reverse triangle inequality: $, p(x) - p(y), \leq p(x - y)$
* For any $r > 0$, $x + \ = \$
* For any $r > 0$, $\$ is an absorbing disk in $X$
* $p(0) = 0$
* $0 \leq \max \$ and $p(x) - p(y) \leq p(x - y)$
* If $p$ is a sublinear function on a real vector space $X$ then there exists a linear functional $f$ on $X$ such that $f \leq p$
* If $X$ is a real vector space, $f$ is a linear functional on $X,$ and $p$ is a sublinear function on $X,$ then $f \leq p$ on $X$ if and only if $f^(1) \cap \$

Other properties of seminorms

Every seminorm is a balanced function.

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