Editing Seminorm (section)

==Topologies of seminormed spaces==

===Pseudometrics and the induced topology===

A seminorm <math>p</math> on <math>X</math> induces a topology, called the {{em|seminorm-induced topology}}, via the canonical [[translation-invariant]] [[Pseudometric space|pseudometric]] <math>d_p : X \times X \to \R</math>; <math>d_p(x, y) := p(x - y) = p(y - x).</math> 
This topology is [[Hausdorff space|Hausdorff]] if and only if <math>d_p</math> is a metric, which occurs if and only if <math>p</math> is a [[Norm (mathematics)|norm]].{{sfn|Wilansky|2013 |pp=15-21}} 
This topology makes <math>X</math> into a [[Locally convex topological vector space|locally convex]] [[Metrizable topological vector space|pseudometrizable]] [[topological vector space]] that has a [[Bounded set (topological vector space)|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:
<math display=block>\{x \in X : p(x) < r\} \quad \text{ or } \quad \{x \in X : p(x) \leq r\}</math>
as <math>r > 0</math> ranges over the positive reals. 
Every seminormed space <math>(X, p)</math> 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 {{em|seminormable}}. 

Equivalently, every vector space <math>X</math> with seminorm <math>p</math> induces a [[Quotient space (linear algebra)|vector space quotient]] <math>X / W,</math> where <math>W</math> is the subspace of <math>X</math> consisting of all vectors <math>x \in X</math> with <math>p(x) = 0.</math> Then <math>X / W</math> carries a norm defined by <math>p(x + W) = p(x).</math>  The resulting topology, [[Pullback|pulled back]] to <math>X,</math> is precisely the topology induced by <math>p.</math>

Any seminorm-induced topology makes <math>X</math> [[Locally convex topological vector space|locally convex]], as follows.  If <math>p</math> is a seminorm on <math>X</math> and <math>r \in \R,</math> call the set <math>\{x \in X : p(x) < r\}</math> the {{em|open ball of radius <math>r</math> about the origin}}; likewise the closed ball of radius <math>r</math> is <math>\{x \in X : p(x) \leq r\}.</math> The set of all open (resp. closed) <math>p</math>-balls at the origin forms a neighborhood basis of [[Convex set|convex]] [[Balanced set|balanced]] sets that are open (resp. closed) in the <math>p</math>-topology on <math>X.</math>

====Stronger, weaker, and equivalent seminorms====

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker [[Norm (mathematics)|norms]].  If <math>p</math> and <math>q</math> are seminorms on <math>X,</math> then we say that <math>q</math> is {{em|stronger}} than <math>p</math> and that <math>p</math> is {{em|weaker}} than <math>q</math> if any of the following equivalent conditions holds:

# The topology on <math>X</math> induced by <math>q</math> is finer than the topology induced by <math>p.</math>
# If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> is a sequence in <math>X,</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0</math> in <math>\R</math> implies <math>p\left(x_{\bull}\right) \to 0</math> in <math>\R.</math>{{sfn|Wilansky|2013 |pp=15-21}}
# If <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a [[Net (mathematics)|net]] in <math>X,</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i \in I} \to 0</math> in <math>\R</math> implies <math>p\left(x_{\bull}\right) \to 0</math> in <math>\R.</math>
# <math>p</math> is bounded on <math>\{x \in X : q(x) < 1\}.</math>{{sfn|Wilansky|2013 |pp=15-21}}
# If <math>\inf{} \{q(x) : p(x) = 1, x \in X\} = 0</math> then <math>p(x) = 0</math> for all <math>x \in X.</math>{{sfn|Wilansky|2013 |pp=15-21}}
# There exists a real <math>K > 0</math> such that <math>p \leq K q</math> on <math>X.</math>{{sfn|Wilansky|2013 |pp=15-21}}

The seminorms <math>p</math> and <math>q</math> are called {{em|equivalent}} if they are both weaker (or both stronger) than each other.  This happens if they satisfy any of the following conditions:
<ol>
<li>The topology on <math>X</math> induced by <math>q</math> is the same as the topology induced by <math>p.</math></li>
<li><math>q</math> is stronger than <math>p</math> and <math>p</math> is stronger than <math>q.</math>{{sfn|Wilansky|2013|pp=15-21}}</li>
<li>If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> is a sequence in <math>X</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0</math> if and only if <math>p\left(x_{\bull}\right) \to 0.</math></li>
<li>There exist positive real numbers <math>r > 0</math> and <math>R > 0</math> such that <math>r q \leq p \leq R q.</math></li>
</ol>

===Normability and seminormability===
{{See also|Normed space|Local boundedness#locally bounded topological vector space}}

A topological vector space (TVS) is said to be a {{em|{{visible anchor|seminormable space}}}} (respectively, a {{em|{{visible anchor|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 space|T<sub>1</sub>]] (because a TVS is Hausdorff if and only if it is a [[T1 space|T<sub>1</sub> space]]). 
A '''{{visible anchor|locally bounded topological vector space}}''' is a topological vector space that possesses a bounded neighborhood of the origin. 

Normability of [[topological vector space]]s is characterized by [[Kolmogorov's normability criterion]]. 
A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.{{sfn|Wilansky|2013|pp=50-51}} 
Thus a [[locally convex]] TVS is seminormable if and only if it has a non-empty bounded open set.{{sfn|Narici|Beckenstein|2011|pp=156-175}}
A TVS is normable if and only if it is a [[T1 space|T<sub>1</sub> space]] and admits a bounded convex neighborhood of the origin.

If <math>X</math> is a Hausdorff [[locally convex]] TVS then the following are equivalent: 
<ol>
<li><math>X</math> is normable.</li>
<li><math>X</math> is seminormable.</li>
<li><math>X</math> has a bounded neighborhood of the origin.</li>
<li>The [[strong dual]] <math>X^{\prime}_b</math> of <math>X</math> is normable.{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}</li>
<li>The strong dual <math>X^{\prime}_b</math> of <math>X</math> is [[Metrizable topological vector space|metrizable]].{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}</li>
</ol>
Furthermore, <math>X</math> is finite dimensional if and only if <math>X^{\prime}_{\sigma}</math> is normable (here <math>X^{\prime}_{\sigma}</math> denotes <math>X^{\prime}</math> 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).{{sfn|Narici|Beckenstein|2011|pp=156–175}}

===Topological properties===

<ul>
<li>If <math>X</math> is a TVS and <math>p</math> is a continuous seminorm on <math>X,</math> then the closure of <math>\{x \in X : p(x) < r\}</math> in <math>X</math> is equal to <math>\{x \in X : p(x) \leq r\}.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li>
<li>The closure of <math>\{0\}</math> in a locally convex space <math>X</math> whose topology is defined by a family of continuous seminorms <math>\mathcal{P}</math> is equal to <math>\bigcap_{p \in \mathcal{P}} p^{-1}(0).</math>{{sfn|Narici|Beckenstein|2011|pp=149-153}}</li>
<li>A subset <math>S</math> in a seminormed space <math>(X, p)</math> is [[Bounded set (topological vector space)|bounded]] if and only if <math>p(S)</math> is bounded.{{sfn|Wilansky|2013|pp=49-50}}</li>
<li>If <math>(X, p)</math> is a seminormed space then the locally convex topology that <math>p</math> induces on <math>X</math> makes <math>X</math> into a [[Metrizable topological vector space|pseudometrizable TVS]] with a canonical pseudometric given by <math>d(x, y) := p(x - y)</math> for all <math>x, y \in X.</math>{{sfn|Narici|Beckenstein|2011|pp=115-154}}</li>
<li>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).{{sfn|Narici|Beckenstein|2011|pp=156–175}}</li>
</ul>

===Continuity of seminorms===

If <math>p</math> is a seminorm on a topological vector space <math>X,</math> then the following are equivalent:{{sfn|Schaefer|Wolff|1999|p=40}} 
<ol>
<li><math>p</math> is continuous.</li>
<li><math>p</math> is continuous at 0;{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li>
<li><math>\{x \in X : p(x) < 1\}</math> is open in <math>X</math>;{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li>
<li><math>\{x \in X : p(x) \leq 1\}</math> is closed neighborhood of 0 in <math>X</math>;{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li>
<li><math>p</math> is uniformly continuous on <math>X</math>;{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li>
<li>There exists a continuous seminorm <math>q</math> on <math>X</math> such that <math>p \leq q.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li>
</ol>

In particular, if <math>(X, p)</math> is a seminormed space then a seminorm <math>q</math> on <math>X</math> is continuous if and only if <math>q</math> is dominated by a positive scalar multiple of <math>p.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}

If <math>X</math> is a real TVS, <math>f</math> is a linear functional on <math>X,</math> and <math>p</math> is a continuous seminorm (or more generally, a sublinear function) on <math>X,</math> then <math>f \leq p</math> on <math>X</math> implies that <math>f</math> is continuous.{{sfn|Narici|Beckenstein|2011|pp=177-220}}

===Continuity of linear maps===

If <math>F : (X, p) \to (Y, q)</math> is a map between seminormed spaces then let{{sfn|Wilansky|2013|pp=21-26}}
<math display="block">\|F\|_{p,q} := \sup \{q(F(x)) : p(x) \leq 1, x \in X\}.</math>

If <math>F : (X, p) \to (Y, q)</math> is a linear map between seminormed spaces then the following are equivalent:
<ol>
<li><math>F</math> is continuous;</li>
<li><math>\|F\|_{p,q} < \infty</math>;{{sfn|Wilansky|2013|pp=21-26}}</li>
<li>There exists a real <math>K \geq 0</math> such that <math>p \leq K q</math>;{{sfn|Wilansky|2013|pp=21-26}}
* In this case, <math>\|F\|_{p,q} \leq K.</math></li>
</ol>
If <math>F</math> is continuous then <math>q(F(x)) \leq \|F\|_{p,q} p(x)</math> for all <math>x \in X.</math>{{sfn|Wilansky|2013|pp=21-26}}

The space of all continuous linear maps <math>F : (X, p) \to (Y, q)</math> between seminormed spaces is itself a seminormed space under the seminorm <math>\|F\|_{p,q}.</math> 
This seminorm is a norm if <math>q</math> is a norm.{{sfn|Wilansky|2013|pp=21-26}}