# Seminorm

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

1. Subadditivity/Triangle inequality: $\displaystyle{ p(x + y) \leq p(x) + p(y) }$ for all $\displaystyle{ x, y \in X. }$
2. Absolute homogeneity: $\displaystyle{ p(s x) =|s|p(x) }$ for all $\displaystyle{ x \in X }$ and all scalars $\displaystyle{ s. }$

These two conditions imply that $\displaystyle{ p(0) = 0 }$[proof 1] and that every seminorm $\displaystyle{ p }$ also has the following property:[proof 2]

1. Nonnegativity: $\displaystyle{ p(x) \geq 0 }$ for all $\displaystyle{ x \in X. }$

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

1. Positive definite/Point-separating: for all $\displaystyle{ x \in X, }$ if $\displaystyle{ p(x) = 0 }$ then $\displaystyle{ x = 0. }$

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 $\displaystyle{ p : X \to \R }$ 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 $\displaystyle{ p : X \to \R }$ is a seminorm if and only if it is a sublinear and balanced function.

## Examples

• The trivial seminorm on $\displaystyle{ X, }$ which refers to the constant $\displaystyle{ 0 }$ map on $\displaystyle{ X, }$ induces the indiscrete topology on $\displaystyle{ X. }$
• If $\displaystyle{ f }$ is any linear form on a vector space then its absolute value $\displaystyle{ |f|, }$ defined by $\displaystyle{ x \mapsto |f(x)|, }$ is a seminorm.
• A sublinear function $\displaystyle{ f : X \to \R }$ on a real vector space $\displaystyle{ X }$ is a seminorm if and only if it is a symmetric function, meaning that $\displaystyle{ f(-x) = f(x) }$ for all $\displaystyle{ x \in X. }$
• Every real-valued sublinear function $\displaystyle{ f : X \to \R }$ on a real vector space $\displaystyle{ X }$ induces a seminorm $\displaystyle{ p : X \to \R }$ defined by $\displaystyle{ p(x) := \max \{f(x), f(-x)\}. }$[1]
• 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 $\displaystyle{ p : X \to \R }$ and $\displaystyle{ q : Y \to \R }$ are seminorms (respectively, norms) on $\displaystyle{ X }$ and $\displaystyle{ Y }$ then the map $\displaystyle{ r : X \times Y \to \R }$ defined by $\displaystyle{ r(x, y) = p(x) + q(y) }$ is a seminorm (respectively, a norm) on $\displaystyle{ X \times Y. }$ In particular, the maps on $\displaystyle{ X \times Y }$ defined by $\displaystyle{ (x, y) \mapsto p(x) }$ and $\displaystyle{ (x, y) \mapsto q(y) }$ are both seminorms on $\displaystyle{ X \times Y. }$
• If $\displaystyle{ p }$ and $\displaystyle{ q }$ are seminorms on $\displaystyle{ X }$ then so are[2] $\displaystyle{ (p \vee q)(x) = \max \{p(x), q(x)\} \quad \text{ and } \quad (p \wedge q)(x) := \inf \{p(y) + q(z) : x = y + z \text{ with } y, z \in X\} }$ where $\displaystyle{ p \wedge q \leq p }$ and $\displaystyle{ p \wedge q \leq q. }$[3]
• The space of seminorms on $\displaystyle{ X }$ is generally not a distributive lattice with respect to the above operations. For example, over $\displaystyle{ \R^2 }$, $\displaystyle{ p(x, y) := \max(|x|, |y|), q(x, y) := 2|x|, r(x, y) := 2|y| }$ are such that$\displaystyle{ ((p \vee q) \wedge (p \vee r)) (x, y) = \inf \{\max(2|x_1|, |y_1|) + \max(|x_2|, 2|y_2|) : x = x_1 + x_2 \text{ and } y = y_1 + y_2\} \quad \text{ while } \quad (p \vee q \wedge r) (x, y) := \max(|x|, |y|) }$
• If $\displaystyle{ L : X \to Y }$ is a linear map and $\displaystyle{ q : Y \to \R }$ is a seminorm on $\displaystyle{ Y, }$ then $\displaystyle{ q \circ L : X \to \R }$ is a seminorm on $\displaystyle{ X. }$ The seminorm $\displaystyle{ q \circ L }$ will be a norm on $\displaystyle{ X }$ if and only if $\displaystyle{ L }$ is injective and the restriction $\displaystyle{ q\big\vert_{L(X)} }$ is a norm on $\displaystyle{ L(X). }$

## Minkowski functionals and seminorms

Main page: Minkowski functional

Seminorms on a vector space $\displaystyle{ X }$ are intimately tied, via Minkowski functionals, to subsets of $\displaystyle{ X }$ that are convex, balanced, and absorbing. Given such a subset $\displaystyle{ D }$ of $\displaystyle{ X, }$ the Minkowski functional of $\displaystyle{ D }$ is a seminorm. Conversely, given a seminorm $\displaystyle{ p }$ on $\displaystyle{ X, }$ the sets$\displaystyle{ \{x \in X : p(x) \lt 1\} }$ and $\displaystyle{ \{x \in X : p(x) \leq 1\} }$ 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 $\displaystyle{ p. }$[4]

## Algebraic properties

Let $\displaystyle{ X }$ be a vector space over $\displaystyle{ \mathbb{F} }$ where $\displaystyle{ \mathbb{F} }$ is either the real or complex numbers.

Properties of seminorms because they are sublinear functions

Since every seminorm is a sublinear function, seminorms have all of the following properties:

If $\displaystyle{ p : X \to [0, \infty) }$ is a real-valued sublinear function on $\displaystyle{ X }$ then:

• Seminorms satisfy the reverse triangle inequality: $\displaystyle{ |p(x) - p(y)| \leq p(x - y) \text{ for all } x, y \in X. }$[5][6]
• For any $\displaystyle{ x \in X }$ and $\displaystyle{ r \gt 0, }$[7] $\displaystyle{ x + \{y \in X : p(y) \lt r\} = \{y \in X : p(x - y) \lt r\}. }$
• Since every seminorm is a sublinear function, every seminorm $\displaystyle{ p }$ on $\displaystyle{ X }$ is a convex function. Moreover, for all $\displaystyle{ r \gt 0, }$ $\displaystyle{ \{x \in X : p(x) \lt r\} }$ is an absorbing disk in $\displaystyle{ X. }$[2]
• Every sublinear function is a convex functional.
• $\displaystyle{ p(0) = 0. }$
• $\displaystyle{ 0 \leq \max \{p(x), p(-x)\} }$ and $\displaystyle{ p(x) - p(y) \leq p(x - y) }$ for all $\displaystyle{ x, y \in X. }$[5][6]
• If $\displaystyle{ p }$ is a sublinear function on a real vector space $\displaystyle{ X }$ then there exists a linear functional $\displaystyle{ f }$ on $\displaystyle{ X }$ such that $\displaystyle{ f \leq p. }$[6]
• If $\displaystyle{ X }$ is a real vector space, $\displaystyle{ f }$ is a linear functional on $\displaystyle{ X, }$ and $\displaystyle{ p }$ is a sublinear function on $\displaystyle{ X, }$ then $\displaystyle{ f \leq p }$ on $\displaystyle{ X }$ if and only if $\displaystyle{ f^{-1}(1) \cap \{x \in X : p(x) \lt 1 = \varnothing\}. }$[6]

Other properties of seminorms

Every seminorm is a balanced function.

If $\displaystyle{ p : X \to [0, \infty) }$ is a seminorm on $\displaystyle{ X }$ then:

• $\displaystyle{ p }$ is a norm on $\displaystyle{ X }$ if and only if$\displaystyle{ \{x \in X : p(x) \lt 1\} }$ does not contain a non-trivial vector subspace.
• $\displaystyle{ p^{-1}(0) }$ is a vector subspace of $\displaystyle{ X. }$
• For any $\displaystyle{ r \gt 0, }$[2] $\displaystyle{ r \{x \in X : p(x) \lt 1\} = \{x \in X : p(x) \lt r\} = \left\{x \in X : \frac{1}{r} p(x) \lt 1 \right\}. }$
• If $\displaystyle{ D }$ is a set satisfying $\displaystyle{ \{x \in X : p(x) \lt 1\} \subseteq D \subseteq \{x \in X : p(x) \leq 1\} }$ then $\displaystyle{ D }$ is absorbing in $\displaystyle{ X }$ and $\displaystyle{ p = p_D }$ where $\displaystyle{ p_D }$ denotes the Minkowski functional associated with $\displaystyle{ D }$ (that is, the gauge of $\displaystyle{ D }$).[4]
• In particular, if $\displaystyle{ D }$ is as above and $\displaystyle{ q }$ is any seminorm on $\displaystyle{ X, }$ then $\displaystyle{ q = p }$ if and only if $\displaystyle{ \{x \in X : q(x) \lt 1\} \subseteq D \subseteq \{x \in X : q(x) \leq\}. }$[4]
• If $\displaystyle{ (X, \|\,\cdot\,\|) }$ is a normed space and $\displaystyle{ x, y \in X }$ then $\displaystyle{ \|x - y\| = \|x - z\| + \|z - y\| }$ for all $\displaystyle{ z \in [x, y]. }$[8]
• Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.

### Relationship to other norm-like concepts

Let $\displaystyle{ p : X \to \R }$ be a non-negative function. The following are equivalent:

1. $\displaystyle{ p }$ is a seminorm.
2. $\displaystyle{ p }$ is a convex $\displaystyle{ F }$-seminorm.
3. $\displaystyle{ p }$ is a convex balanced G-seminorm.[9]

If any of the above conditions hold, then the following are equivalent:

1. $\displaystyle{ p }$ is a norm;
2. $\displaystyle{ \{x \in X : p(x) \lt 1\} }$ does not contain a non-trivial vector subspace.[10]
3. There exists a norm on $\displaystyle{ X, }$ with respect to which, $\displaystyle{ \{x \in X : p(x) \lt 1\} }$ is bounded.

If $\displaystyle{ p }$ is a sublinear function on a real vector space $\displaystyle{ X }$ then the following are equivalent:[6]

1. $\displaystyle{ p }$ is a linear functional;
2. $\displaystyle{ p(x) + p(-x) \leq 0 \text{ for every } x \in X }$;
3. $\displaystyle{ p(x) + p(-x) = 0 \text{ for every } x \in X }$;

### Inequalities involving seminorms

If $\displaystyle{ p, q : X \to [0, \infty) }$ are seminorms on $\displaystyle{ X }$ then:

• $\displaystyle{ p \leq q }$ if and only if $\displaystyle{ q(x) \leq 1 }$ implies $\displaystyle{ p(x) \leq 1. }$[11]
• If $\displaystyle{ a \gt 0 }$ and $\displaystyle{ b \gt 0 }$ are such that $\displaystyle{ p(x) \lt a }$ implies $\displaystyle{ q(x) \leq b, }$ then $\displaystyle{ a q(x) \leq b p(x) }$ for all $\displaystyle{ x \in X. }$ [12]
• Suppose $\displaystyle{ a }$ and $\displaystyle{ b }$ are positive real numbers and $\displaystyle{ q, p_1, \ldots, p_n }$ are seminorms on $\displaystyle{ X }$ such that for every $\displaystyle{ x \in X, }$ if $\displaystyle{ \max \{p_1(x), \ldots, p_n(x)\} \lt a }$ then $\displaystyle{ q(x) \lt b. }$ Then $\displaystyle{ a q \leq b \left(p_1 + \cdots + p_n\right). }$[10]
• If $\displaystyle{ X }$ is a vector space over the reals and $\displaystyle{ f }$ is a non-zero linear functional on $\displaystyle{ X, }$ then $\displaystyle{ f \leq p }$ if and only if $\displaystyle{ \varnothing = f^{-1}(1) \cap \{x \in X : p(x) \lt 1\}. }$[11]

If $\displaystyle{ p }$ is a seminorm on $\displaystyle{ X }$ and $\displaystyle{ f }$ is a linear functional on $\displaystyle{ X }$ then:

• $\displaystyle{ |f| \leq p }$ on $\displaystyle{ X }$ if and only if $\displaystyle{ \operatorname{Re} f \leq p }$ on $\displaystyle{ X }$ (see footnote for proof).[13][14]
• $\displaystyle{ f \leq p }$ on $\displaystyle{ X }$ if and only if $\displaystyle{ f^{-1}(1) \cap \{x \in X : p(x) \lt 1 = \varnothing\}. }$[6][11]
• If $\displaystyle{ a \gt 0 }$ and $\displaystyle{ b \gt 0 }$ are such that $\displaystyle{ p(x) \lt a }$ implies $\displaystyle{ f(x) \neq b, }$ then $\displaystyle{ a |f(x)| \leq b p(x) }$ for all $\displaystyle{ x \in X. }$[12]

### Hahn–Banach theorem for seminorms

Seminorms offer a particularly clean formulation of the Hahn–Banach theorem:

If $\displaystyle{ M }$ is a vector subspace of a seminormed space $\displaystyle{ (X, p) }$ and if $\displaystyle{ f }$ is a continuous linear functional on $\displaystyle{ M, }$ then $\displaystyle{ f }$ may be extended to a continuous linear functional $\displaystyle{ F }$ on $\displaystyle{ X }$ that has the same norm as $\displaystyle{ f. }$[15]

A similar extension property also holds for seminorms:

Theorem[16][12] (Extending seminorms) — If $\displaystyle{ M }$ is a vector subspace of $\displaystyle{ X, }$ $\displaystyle{ p }$ is a seminorm on $\displaystyle{ M, }$ and $\displaystyle{ q }$ is a seminorm on $\displaystyle{ X }$ such that $\displaystyle{ p \leq q\big\vert_M, }$ then there exists a seminorm $\displaystyle{ P }$ on $\displaystyle{ X }$ such that $\displaystyle{ P\big\vert_M = p }$ and $\displaystyle{ P \leq q. }$

Proof: Let $\displaystyle{ S }$ be the convex hull of $\displaystyle{ \{m \in M : p(m) \leq 1\} \cup \{x \in X : q(x) \leq 1\}. }$ Then $\displaystyle{ S }$ is an absorbing disk in $\displaystyle{ X }$ and so the Minkowski functional $\displaystyle{ P }$ of $\displaystyle{ S }$ is a seminorm on $\displaystyle{ X. }$ This seminorm satisfies $\displaystyle{ p = P }$ on $\displaystyle{ M }$ and $\displaystyle{ P \leq q }$ on $\displaystyle{ X. }$ $\displaystyle{ \blacksquare }$

## Topologies of seminormed spaces

### Pseudometrics and the induced topology

A seminorm $\displaystyle{ p }$ on $\displaystyle{ X }$ induces a topology, called the seminorm-induced topology, via the canonical translation-invariant pseudometric $\displaystyle{ d_p : X \times X \to \R }$; $\displaystyle{ d_p(x, y) := p(x - y) = p(y - x). }$ This topology is Hausdorff if and only if $\displaystyle{ d_p }$ is a metric, which occurs if and only if $\displaystyle{ p }$ is a norm.[3] This topology makes $\displaystyle{ X }$ into a locally convex pseudometrizable topological vector space that has a bounded neighborhood of the origin and a neighborhood basis at the origin consisting of the following open balls (or the closed balls) centered at the origin: $\displaystyle{ \{x \in X : p(x) \lt r\} \quad \text{ or } \quad \{x \in X : p(x) \leq r\} }$ as $\displaystyle{ r \gt 0 }$ ranges over the positive reals. Every seminormed space $\displaystyle{ (X, p) }$ should be assumed to be endowed with this topology unless indicated otherwise. A topological vector space whose topology is induced by some seminorm is called seminormable.

Equivalently, every vector space $\displaystyle{ X }$ with seminorm $\displaystyle{ p }$ induces a vector space quotient $\displaystyle{ X / W, }$ where $\displaystyle{ W }$ is the subspace of $\displaystyle{ X }$ consisting of all vectors $\displaystyle{ x \in X }$ with $\displaystyle{ p(x) = 0. }$ Then $\displaystyle{ X / W }$ carries a norm defined by $\displaystyle{ p(x + W) = p(v). }$ The resulting topology, pulled back to $\displaystyle{ X, }$ is precisely the topology induced by $\displaystyle{ p. }$

Any seminorm-induced topology makes $\displaystyle{ X }$ locally convex, as follows. If $\displaystyle{ p }$ is a seminorm on $\displaystyle{ X }$ and $\displaystyle{ r \in \R, }$ call the set $\displaystyle{ \{x \in X : p(x) \lt r\} }$ the open ball of radius $\displaystyle{ r }$ about the origin; likewise the closed ball of radius $\displaystyle{ r }$ is $\displaystyle{ \{x \in X : p(x) \leq r\}. }$ The set of all open (resp. closed) $\displaystyle{ p }$-balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the $\displaystyle{ p }$-topology on $\displaystyle{ X. }$

#### Stronger, weaker, and equivalent seminorms

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If $\displaystyle{ p }$ and $\displaystyle{ q }$ are seminorms on $\displaystyle{ X, }$ then we say that $\displaystyle{ q }$ is stronger than $\displaystyle{ p }$ and that $\displaystyle{ p }$ is weaker than $\displaystyle{ q }$ if any of the following equivalent conditions holds:

1. The topology on $\displaystyle{ X }$ induced by $\displaystyle{ q }$ is finer than the topology induced by $\displaystyle{ p. }$
2. If $\displaystyle{ x_{\bull} = \left(x_i\right)_{i=1}^{\infty} }$ is a sequence in $\displaystyle{ X, }$ then $\displaystyle{ q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0 }$ in $\displaystyle{ \R }$ implies $\displaystyle{ p\left(x_{\bull}\right) \to 0 }$ in $\displaystyle{ \R. }$[3]
3. If $\displaystyle{ x_{\bull} = \left(x_i\right)_{i \in I} }$ is a net in $\displaystyle{ X, }$ then $\displaystyle{ q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i \in I} \to 0 }$ in $\displaystyle{ \R }$ implies $\displaystyle{ p\left(x_{\bull}\right) \to 0 }$ in $\displaystyle{ \R. }$
4. $\displaystyle{ p }$ is bounded on $\displaystyle{ \{x \in X : q(x) \lt 1\}. }$[3]
5. If $\displaystyle{ \inf{} \{q(x) : p(x) = 1, x \in X\} = 0 }$ then $\displaystyle{ p(x) = 0 }$ for all $\displaystyle{ x \in X. }$[3]
6. There exists a real $\displaystyle{ K \gt 0 }$ such that $\displaystyle{ p \leq K q }$ on $\displaystyle{ X. }$[3]

The seminorms $\displaystyle{ p }$ and $\displaystyle{ q }$ are called equivalent if they are both weaker (or both stronger) than each other. This happens if they satisfy any of the following conditions:

1. The topology on $\displaystyle{ X }$ induced by $\displaystyle{ q }$ is the same as the topology induced by $\displaystyle{ p. }$
2. $\displaystyle{ q }$ is stronger than $\displaystyle{ p }$ and $\displaystyle{ p }$ is stronger than $\displaystyle{ q. }$[3]
3. If $\displaystyle{ x_{\bull} = \left(x_i\right)_{i=1}^{\infty} }$ is a sequence in $\displaystyle{ X }$ then $\displaystyle{ q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0 }$ if and only if $\displaystyle{ p\left(x_{\bull}\right) \to 0. }$
4. There exist positive real numbers $\displaystyle{ r \gt 0 }$ and $\displaystyle{ R \gt 0 }$ such that $\displaystyle{ r q \leq p \leq R q. }$

### Normability and seminormability

A topological vector space (TVS) is said to be a seminormable space (respectively, a normable space) if its topology is induced by a single seminorm (resp. a single norm). A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and T1 (because a TVS is Hausdorff if and only if it is a T1 space). A locally bounded topological vector space is a topological vector space that possesses a bounded neighborhood of the origin.

Normability of topological vector spaces is characterized by Kolmogorov's normability criterion. A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.[17] Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.[18] A TVS is normable if and only if it is a T1 space and admits a bounded convex neighborhood of the origin.

If $\displaystyle{ X }$ is a Hausdorff locally convex TVS then the following are equivalent:

1. $\displaystyle{ X }$ is normable.
2. $\displaystyle{ X }$ is seminormable.
3. $\displaystyle{ X }$ has a bounded neighborhood of the origin.
4. The strong dual $\displaystyle{ X^{\prime}_b }$ of $\displaystyle{ X }$ is normable.[19]
5. The strong dual $\displaystyle{ X^{\prime}_b }$ of $\displaystyle{ X }$ is metrizable.[19]

Furthermore, $\displaystyle{ X }$ is finite dimensional if and only if $\displaystyle{ X^{\prime}_{\sigma} }$ is normable (here $\displaystyle{ X^{\prime}_{\sigma} }$ denotes $\displaystyle{ X^{\prime} }$ endowed with the weak-* topology).

The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial (that is, 0-dimensional).[20]

### Topological properties

• If $\displaystyle{ X }$ is a TVS and $\displaystyle{ p }$ is a continuous seminorm on $\displaystyle{ X, }$ then the closure of $\displaystyle{ \{x \in X : p(x) \lt r\} }$ in $\displaystyle{ X }$ is equal to $\displaystyle{ \{x \in X : p(x) \leq r\}. }$[2]
• The closure of $\displaystyle{ \{0\} }$ in a locally convex space $\displaystyle{ X }$ whose topology is defined by a family of continuous seminorms $\displaystyle{ \mathcal{P} }$ is equal to $\displaystyle{ \bigcap_{p \in \mathcal{P}} p^{-1}(0). }$[21]
• A subset $\displaystyle{ S }$ in a seminormed space $\displaystyle{ (X, p) }$ is bounded if and only if $\displaystyle{ p(S) }$ is bounded.[22]
• If $\displaystyle{ (X, p) }$ is a seminormed space then the locally convex topology that $\displaystyle{ p }$ induces on $\displaystyle{ X }$ makes $\displaystyle{ X }$ into a pseudometrizable TVS with a canonical pseudometric given by $\displaystyle{ d(x, y) := p(x - y) }$ for all $\displaystyle{ x, y \in X. }$[23]
• The product of infinitely many seminormable spaces is again seminormable if and only if all but finitely many of these spaces are trivial (that is, 0-dimensional).[20]

### Continuity of seminorms

If $\displaystyle{ p }$ is a seminorm on a topological vector space $\displaystyle{ X, }$ then the following are equivalent:[4]

1. $\displaystyle{ p }$ is continuous.
2. $\displaystyle{ p }$ is continuous at 0;[2]
3. $\displaystyle{ \{x \in X : p(x) \lt 1\} }$ is open in $\displaystyle{ X }$;[2]
4. $\displaystyle{ \{x \in X : p(x) \leq 1\} }$ is closed neighborhood of 0 in $\displaystyle{ X }$;[2]
5. $\displaystyle{ p }$ is uniformly continuous on $\displaystyle{ X }$;[2]
6. There exists a continuous seminorm $\displaystyle{ q }$ on $\displaystyle{ X }$ such that $\displaystyle{ p \leq q. }$[2]

In particular, if $\displaystyle{ (X, p) }$ is a seminormed space then a seminorm $\displaystyle{ q }$ on $\displaystyle{ X }$ is continuous if and only if $\displaystyle{ q }$ is dominated by a positive scalar multiple of $\displaystyle{ p. }$[2]

If $\displaystyle{ X }$ is a real TVS, $\displaystyle{ f }$ is a linear functional on $\displaystyle{ X, }$ and $\displaystyle{ p }$ is a continuous seminorm (or more generally, a sublinear function) on $\displaystyle{ X, }$ then $\displaystyle{ f \leq p }$ on $\displaystyle{ X }$ implies that $\displaystyle{ f }$ is continuous.[6]

### Continuity of linear maps

If $\displaystyle{ F : (X, p) \to (Y, q) }$ is a map between seminormed spaces then let[15] $\displaystyle{ \|F\|_{p,q} := \sup \{q(F(x)) : p(x) \leq 1, x \in X\}. }$

If $\displaystyle{ F : (X, p) \to (Y, q) }$ is a linear map between seminormed spaces then the following are equivalent:

1. $\displaystyle{ F }$ is continuous;
2. $\displaystyle{ \|F\|_{p,q} \lt \infty }$;[15]
3. There exists a real $\displaystyle{ K \geq 0 }$ such that $\displaystyle{ p \leq K q }$;[15]
• In this case, $\displaystyle{ \|F\|_{p,q} \leq K. }$

If $\displaystyle{ F }$ is continuous then $\displaystyle{ q(F(x)) \leq \|F\|_{p,q} p(x) }$ for all $\displaystyle{ x \in X. }$[15]

The space of all continuous linear maps $\displaystyle{ F : (X, p) \to (Y, q) }$ between seminormed spaces is itself a seminormed space under the seminorm $\displaystyle{ \|F\|_{p,q}. }$ This seminorm is a norm if $\displaystyle{ q }$ is a norm.[15]

## Generalizations

The concept of norm in composition algebras does not share the usual properties of a norm.

A composition algebra $\displaystyle{ (A, *, N) }$ consists of an algebra over a field $\displaystyle{ A, }$ an involution $\displaystyle{ \,*, }$ and a quadratic form $\displaystyle{ N, }$ which is called the "norm". In several cases $\displaystyle{ N }$ is an isotropic quadratic form so that $\displaystyle{ A }$ has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.

An ultraseminorm or a non-Archimedean seminorm is a seminorm $\displaystyle{ p : X \to \R }$ that also satisfies $\displaystyle{ p(x + y) \leq \max \{p(x), p(y)\} \text{ for all } x, y \in X. }$

A map $\displaystyle{ p : X \to \R }$ is called a quasi-seminorm if it is (absolutely) homogeneous and there exists some $\displaystyle{ b \leq 1 }$ such that $\displaystyle{ p(x + y) \leq b p(p(x) + p(y)) \text{ for all } x, y \in X. }$ The smallest value of $\displaystyle{ b }$ for which this holds is called the multiplier of $\displaystyle{ p. }$

A quasi-seminorm that separates points is called a quasi-norm on $\displaystyle{ X. }$

Weakening homogeneity - $\displaystyle{ k }$-seminorms

A map $\displaystyle{ p : X \to \R }$ is called a $\displaystyle{ k }$-seminorm if it is subadditive and there exists a $\displaystyle{ k }$ such that $\displaystyle{ 0 \lt k \leq 1 }$ and for all $\displaystyle{ x \in X }$ and scalars $\displaystyle{ s, }$$\displaystyle{ p(s x) = |s|^k p(x) }$ A $\displaystyle{ k }$-seminorm that separates points is called a $\displaystyle{ k }$-norm on $\displaystyle{ X. }$

We have the following relationship between quasi-seminorms and $\displaystyle{ k }$-seminorms:

Suppose that $\displaystyle{ q }$ is a quasi-seminorm on a vector space $\displaystyle{ X }$ with multiplier $\displaystyle{ b. }$ If $\displaystyle{ 0 \lt \sqrt{k} \lt \log_2 b }$ then there exists $\displaystyle{ k }$-seminorm $\displaystyle{ p }$ on $\displaystyle{ X }$ equivalent to $\displaystyle{ q. }$

## Notes

Proofs

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

## References

1. Narici & Beckenstein 2011, pp. 120–121.
2. Narici & Beckenstein 2011, pp. 116–128.
3. Wilansky 2013, pp. 15-21.
4. Schaefer & Wolff 1999, p. 40.
5. Narici & Beckenstein 2011, pp. 120-121.
6. Narici & Beckenstein 2011, pp. 177-220.
7. Narici & Beckenstein 2011, pp. 116−128.
8. Narici & Beckenstein 2011, pp. 107-113.
9. Schechter 1996, p. 691.
10. Narici & Beckenstein 2011, p. 149.
11. Narici & Beckenstein 2011, pp. 149–153.
12. Wilansky 2013, pp. 18-21.
13. Obvious if $\displaystyle{ X }$ is a real vector space. For the non-trivial direction, assume that $\displaystyle{ \operatorname{Re} f \leq p }$ on $\displaystyle{ X }$ and let $\displaystyle{ x \in X. }$ Let $\displaystyle{ r \geq 0 }$ and $\displaystyle{ t }$ be real numbers such that $\displaystyle{ f(x) = r e^{i t}. }$ Then $\displaystyle{ |f(x)|= r = f\left(e^{-it} x\right) = \operatorname{Re}\left(f\left(e^{-it} x\right)\right) \leq p\left(e^{-it} x\right) = p(x). }$
14. Wilansky 2013, p. 20.
15. Wilansky 2013, pp. 21-26.
16. Narici & Beckenstein 2011, pp. 150.
17. Wilansky 2013, pp. 50-51.
18. Narici & Beckenstein 2011, pp. 156-175.
19. Trèves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433.
20. Narici & Beckenstein 2011, pp. 156–175.
21. Narici & Beckenstein 2011, pp. 149-153.
22. Wilansky 2013, pp. 49-50.
23. Narici & Beckenstein 2011, pp. 115-154.