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In `F33f`_`[linear algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_algebra]`_`f, a `!sublinear`! function (or `F33f`_`[functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Functional_(mathematics)]`_`f as is more often used in `F33f`_`[functional analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Functional_analysis]`_`f), also called a `!quasi-seminorm`! or a `!Banach functional`!, on a `F33f`_`[vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_space]`_`f X {\\displaystyle X} is a `F33f`_`[real`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f-valued `F33f`_`[function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f with only some of the properties of a `F33f`_`[seminorm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Seminorm]`_`f. Unlike seminorms, a sublinear function does not have to be `F33f`_`[nonnegative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nonnegative]`_`f-valued and also does not have to be `F33f`_`[absolutely homogeneous`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Absolutely_homogeneous]`_`f. Seminorms are themselves abstractions of the more well known notion of `F33f`_`[norms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Norm_(mathematics)]`_`f, where a seminorm has all the defining properties of a norm `*except`* that it is not required to map non-zero vectors to non-zero values.

In `F33f`_`[functional analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Functional_analysis]`_`f the name `!Banach functional`! is sometimes used, reflecting that they are most commonly used when applying a general formulation of the `F33f`_`[Hahn–Banach theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hahn–Banach_theorem]`_`f. The notion of a sublinear function was introduced by `F33f`_`[Stefan Banach`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stefan_Banach]`_`f when he proved his version of the `F33f`_`[Hahn-Banach theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hahn-Banach_theorem]`_`f.`:cite-ref-footnotenaricibeckenstein2011177-220-1-0[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f]

There is also a different notion in `F33f`_`[computer science`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Computer_science]`_`f, described below, that also goes by the name "sublinear function."

>>Contents

• `F0af`_`[Definitions`#definitions]`_`f

• `F0af`_`[Examples and sufficient conditions`#examples-and-sufficient-conditions]`_`f

• `F0af`_`[Properties`#properties]`_`f

• `F0af`_`[Associated seminorm`#associated-seminorm]`_`f

• `F0af`_`[Relation to linear functionals`#relation-to-linear-functionals]`_`f

• `F0af`_`[Continuity`#continuity]`_`f

• `F0af`_`[Relation to Minkowski functions and open convex sets`#relation-to-minkowski-functions-and-open-convex-sets]`_`f

• `F0af`_`[Operators`#operators]`_`f

• `F0af`_`[Computer science definition`#computer-science-definition]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[Bibliography`#bibliography]`_`f

-─

>>Definitions

Let X {\\displaystyle X} be a `F33f`_`[vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_space]`_`f over a field K , {\\displaystyle \\mathbb {K} ,} where K {\\displaystyle \\mathbb {K} } is either the `F33f`_`[real numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f R {\\displaystyle \\mathbb {R} } or `F33f`_`[complex numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f C . {\\displaystyle \\mathbb {C} .} A real-valued function p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } on X {\\displaystyle X} is called a `*`*sublinear function`*`* (or a `*`*sublinear `F33f`_`[functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Functional_(mathematics)]`_`f`*`* if K = R {\\displaystyle \\mathbb {K} =\\mathbb {R} } ), and also sometimes called a `*`*quasi-seminorm`*`* or a `*`*Banach functional`*`*, if it has these two properties:`:cite-ref-footnotenaricibeckenstein2011177-220-1-1[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f]

1. `*`F33f`_`[Positive homogeneity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Positive_homogeneity]`_`f`!/`!`F33f`_`[Nonnegative homogeneity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nonnegative_homogeneity]`_`f`*:`:cite-ref-footnoteschechter1996313-315-2-0[`F5bf`_`[2`#cite-note-footnoteschechter1996313-315-2]`_`f] p ( r x ) = r p ( x ) {\\displaystyle p(rx)=rp(x)} for all real r ≥ ≥ 0 {\\displaystyle r\\geq 0} and all x ∈ ∈ X . {\\displaystyle x\\in X.}

• This condition holds if and only if p ( r x ) = r p ( x ) {\\displaystyle p(rx)=rp(x)} for all positive real r > 0 {\\displaystyle r>0} and all x ∈ ∈ X . {\\displaystyle x\\in X.}

2. `*`F33f`_`[Subadditivity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Subadditivity]`_`f`!/`!`F33f`_`[Triangle inequality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Triangle_inequality]`_`f`*:`:cite-ref-footnoteschechter1996313-315-2-1[`F5bf`_`[2`#cite-note-footnoteschechter1996313-315-2]`_`f] p ( x + y ) ≤ ≤ p ( x ) + p ( y ) {\\displaystyle p(x+y)\\leq p(x)+p(y)} for all x , y ∈ ∈ X . {\\displaystyle x,y\\in X.}

• This subadditivity condition requires p {\\displaystyle p} to be real-valued.

A function p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } is called `*positive`*`:cite-ref-footnotenaricibeckenstein2011120-121-3-0[`F5bf`_`[3`#cite-note-footnotenaricibeckenstein2011120-121-3]`_`f] or `*nonnegative`* if p ( x ) ≥ ≥ 0 {\\displaystyle p(x)\\geq 0} for all x ∈ ∈ X , {\\displaystyle x\\in X,} although some authors`:cite-ref-footnotekubrusly2011200-4-0[`F5bf`_`[4`#cite-note-footnotekubrusly2011200-4]`_`f] define `*positive`* to instead mean that p ( x ) ≠ ≠ 0 {\\displaystyle p(x)\\neq 0} whenever x ≠ ≠ 0 ; {\\displaystyle x\\neq 0;} these definitions are not equivalent. It is a `*symmetric function`* if p ( − − x ) = p ( x ) {\\displaystyle p(-x)=p(x)} for all x ∈ ∈ X . {\\displaystyle x\\in X.} Every subadditive symmetric function is necessarily nonnegative.`:cite-ref-subadditivesymmetricisnonnegative-5-0[`F5bf`_`[proof 1`#cite-note-subadditivesymmetricisnonnegative-5]`_`f] A sublinear function on a real vector space is `F33f`_`[symmetric`#symmetric-function]`_`f if and only if it is a `F33f`_`[seminorm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Seminorm]`_`f. A sublinear function on a real or complex vector space is a seminorm if and only if it is a `F33f`_`[balanced function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Balanced_function]`_`f or equivalently, if and only if p ( u x ) ≤ ≤ p ( x ) {\\displaystyle p(ux)\\leq p(x)} for every `F33f`_`[unit length`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Unit_length]`_`f scalar u {\\displaystyle u} (satisfying | u | = 1 {\\displaystyle |u|=1} ) and every x ∈ ∈ X . {\\displaystyle x\\in X.}

The set of all sublinear functions on X , {\\displaystyle X,} denoted by X # # , {\\displaystyle X^{\\#},} can be `F33f`_`[partially ordered`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Partial_order]`_`f by declaring p ≤ ≤ q {\\displaystyle p\\leq q} if and only if p ( x ) ≤ ≤ q ( x ) {\\displaystyle p(x)\\leq q(x)} for all x ∈ ∈ X . {\\displaystyle x\\in X.} A sublinear function is called `*`*minimal`*`* if it is a `F33f`_`[minimal element`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Minimal_element]`_`f of X # # {\\displaystyle X^{\\#}} under this order. A sublinear function is minimal if and only if it is a real `F33f`_`[linear functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_functional]`_`f.`:cite-ref-footnotenaricibeckenstein2011177-220-1-2[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f]

>>Examples and sufficient conditions

Every `F33f`_`[norm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Norm_(mathematics)]`_`f, `F33f`_`[seminorm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Seminorm]`_`f, and real linear functional is a sublinear function. The `F33f`_`[identity function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Identity_function]`_`f R → → R {\\displaystyle \\mathbb {R} \\to \\mathbb {R} } on X := R {\\displaystyle X:=\\mathbb {R} } is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation x ↦ ↦ − − x . {\\displaystyle x\\mapsto -x.} `:cite-ref-footnotenaricibeckenstein2011177-221-6-0[`F5bf`_`[5`#cite-note-footnotenaricibeckenstein2011177-221-6]`_`f] More generally, for any real a ≤ ≤ b , {\\displaystyle a\\leq b,} the map S a , b : R → → R x ↦ ↦ { a x if x ≤ ≤ 0 b x if x ≥ ≥ 0 {\\displaystyle {\\begin{alignedat}{4}S_{a,b}:\\;&&\\mathbb {R} &&\\;\\to \\;&\\mathbb {R} \\\\[0.3ex]&&x&&\\;\\mapsto \\;&{\\begin{cases}ax&{\\text{ if }}x\\leq 0\\\\bx&{\\text{ if }}x\\geq 0\\\\\\end{cases}}\\\\\\end{alignedat}}} is a sublinear function on X := R {\\displaystyle X:=\\mathbb {R} } and moreover, every sublinear function p : R → → R {\\displaystyle p:\\mathbb {R} \\to \\mathbb {R} } is of this form; specifically, if a := − − p ( − − 1 ) {\\displaystyle a:=-p(-1)} and b := p ( 1 ) {\\displaystyle b:=p(1)} then a ≤ ≤ b {\\displaystyle a\\leq b} and p = S a , b . {\\displaystyle p=S_{a,b}.}

If p {\\displaystyle p} and q {\\displaystyle q} are sublinear functions on a real vector space X {\\displaystyle X} then so is the map x ↦ ↦ max { p ( x ) , q ( x ) } . {\\displaystyle x\\mapsto \\max\\{p(x),q(x)\\}.} More generally, if P {\\displaystyle {\\mathcal {P}}} is any non-empty collection of sublinear functionals on a real vector space X {\\displaystyle X} and if for all x ∈ ∈ X , {\\displaystyle x\\in X,} q ( x ) := sup { p ( x ) : p ∈ ∈ P } , {\\displaystyle q(x):=\\sup\\{p(x):p\\in {\\mathcal {P}}\\},} then q {\\displaystyle q} is a sublinear functional on X . {\\displaystyle X.} `:cite-ref-footnotenaricibeckenstein2011177-221-6-1[`F5bf`_`[5`#cite-note-footnotenaricibeckenstein2011177-221-6]`_`f]

A function p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } which is `F33f`_`[subadditive`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Subadditivity]`_`f, `F33f`_`[convex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_function]`_`f, and satisfies p ( 0 ) ≤ ≤ 0 {\\displaystyle p(0)\\leq 0} is also positively homogeneous (the latter condition p ( 0 ) ≤ ≤ 0 {\\displaystyle p(0)\\leq 0} is necessary as the example of p ( x ) := x 2 + 1 {\\displaystyle p(x):={\\sqrt {x^{2}+1}}} on X := R {\\displaystyle X:=\\mathbb {R} } shows). If p {\\displaystyle p} is positively homogeneous, it is convex if and only if it is subadditive. Therefore, assuming p ( 0 ) ≤ ≤ 0 {\\displaystyle p(0)\\leq 0} , any two properties among subadditivity, convexity, and positive homogeneity implies the third.

>>Properties

Every sublinear function is a `F33f`_`[convex function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_function]`_`f: For 0 ≤ ≤ t ≤ ≤ 1 , {\\displaystyle 0\\leq t\\leq 1,} p ( t x + ( 1 − − t ) y ) ≤ ≤ p ( t x ) + p ( ( 1 − − t ) y ) subadditivity = t p ( x ) + ( 1 − − t ) p ( y ) nonnegative homogeneity {\\displaystyle {\\begin{alignedat}{3}p(tx+(1-t)y)&\\leq p(tx)+p((1-t)y)&&\\quad {\\text{ subadditivity}}\\\\&=tp(x)+(1-t)p(y)&&\\quad {\\text{ nonnegative homogeneity}}\\\\\\end{alignedat}}}

If p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } is a sublinear function on a vector space X {\\displaystyle X} then`:cite-ref-nullatzeroandsumupperbound-7-0[`F5bf`_`[proof 2`#cite-note-nullatzeroandsumupperbound-7]`_`f]`:cite-ref-footnotenaricibeckenstein2011120-121-3-1[`F5bf`_`[3`#cite-note-footnotenaricibeckenstein2011120-121-3]`_`f] p ( 0 ) = 0 ≤ ≤ p ( x ) + p ( − − x ) , {\\displaystyle p(0)~=~0~\\leq ~p(x)+p(-x),} for every x ∈ ∈ X , {\\displaystyle x\\in X,} which implies that at least one of p ( x ) {\\displaystyle p(x)} and p ( − − x ) {\\displaystyle p(-x)} must be nonnegative; that is, for every x ∈ ∈ X , {\\displaystyle x\\in X,} `:cite-ref-footnotenaricibeckenstein2011120-121-3-2[`F5bf`_`[3`#cite-note-footnotenaricibeckenstein2011120-121-3]`_`f] 0 ≤ ≤ max { p ( x ) , p ( − − x ) } . {\\displaystyle 0~\\leq ~\\max\\{p(x),p(-x)\\}.} Moreover, when p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } is a sublinear function on a real vector space then the map q : X → → R {\\displaystyle q:X\\to \\mathbb {R} } defined by q ( x ) = def max { p ( x ) , p ( − − x ) } {\\displaystyle q(x)~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~\\max\\{p(x),p(-x)\\}} is a seminorm.`:cite-ref-footnotenaricibeckenstein2011120-121-3-3[`F5bf`_`[3`#cite-note-footnotenaricibeckenstein2011120-121-3]`_`f]

Subadditivity of p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } guarantees that for all vectors x , y ∈ ∈ X , {\\displaystyle x,y\\in X,} `:cite-ref-footnotenaricibeckenstein2011177-220-1-3[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f]`:cite-ref-reversetriangle-8-0[`F5bf`_`[proof 3`#cite-note-reversetriangle-8]`_`f] p ( x ) − − p ( y ) ≤ ≤ p ( x − − y ) , {\\displaystyle p(x)-p(y)~\\leq ~p(x-y),} − − p ( x ) ≤ ≤ p ( − − x ) , {\\displaystyle -p(x)~\\leq ~p(-x),} so if p {\\displaystyle p} is also `F33f`_`[symmetric`#symmetric-function]`_`f then the `F33f`_`[reverse triangle inequality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Reverse_triangle_inequality]`_`f will hold for all vectors x , y ∈ ∈ X , {\\displaystyle x,y\\in X,} | p ( x ) − − p ( y ) | ≤ ≤ p ( x − − y ) . {\\displaystyle |p(x)-p(y)|~\\leq ~p(x-y).}

Defining ker ⁡ ⁡ p = def p − − 1 ( 0 ) , {\\displaystyle \\ker p~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~p^{-1}(0),} then subadditivity also guarantees that for all x ∈ ∈ X , {\\displaystyle x\\in X,} the value of p {\\displaystyle p} on the set x + ( ker ⁡ ⁡ p ∩ ∩ − − ker ⁡ ⁡ p ) = { x + k : p ( k ) = 0 = p ( − − k ) } {\\displaystyle x+(\\ker p\\cap -\\ker p)=\\{x+k:p(k)=0=p(-k)\\}} is constant and equal to p ( x ) . {\\displaystyle p(x).} `:cite-ref-constantonequivclasses-9-0[`F5bf`_`[proof 4`#cite-note-constantonequivclasses-9]`_`f] In particular, if ker ⁡ ⁡ p = p − − 1 ( 0 ) {\\displaystyle \\ker p=p^{-1}(0)} is a vector subspace of X {\\displaystyle X} then − − ker ⁡ ⁡ p = ker ⁡ ⁡ p {\\displaystyle -\\ker p=\\ker p} and the assignment x + ker ⁡ ⁡ p ↦ ↦ p ( x ) , {\\displaystyle x+\\ker p\\mapsto p(x),} which will be denoted by p ^ ^ , {\\displaystyle {\\hat {p}},} is a well-defined real-valued sublinear function on the `F33f`_`[quotient space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quotient_space_(linear_algebra)]`_`f X / ker ⁡ ⁡ p {\\displaystyle X\\,/\\,\\ker p} that satisfies p ^ ^ − − 1 ( 0 ) = ker ⁡ ⁡ p . {\\displaystyle {\\hat {p}}^{-1}(0)=\\ker p.} If p {\\displaystyle p} is a seminorm then p ^ ^ {\\displaystyle {\\hat {p}}} is just the usual canonical norm on the quotient space X / ker ⁡ ⁡ p . {\\displaystyle X\\,/\\,\\ker p.}

`!Pryce's sublinearity lemma`:cite-ref-footnoteschechter1996313-315-2-2[`F5bf`_`[2`#cite-note-footnoteschechter1996313-315-2]`_`f]`!—Suppose p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } is a sublinear functional on a vector space X {\\displaystyle X} and that K ⊆ ⊆ X {\\displaystyle K\\subseteq X} is a non-empty convex subset. If x ∈ ∈ X {\\displaystyle x\\in X} is a vector and a , c > 0 {\\displaystyle a,c>0} are positive real numbers such that p ( x ) + a c < inf k ∈ ∈ K p ( x + a k ) {\\displaystyle p(x)+ac~<~\\inf _{k\\in K}p(x+ak)} then for every positive real b > 0 {\\displaystyle b>0} there exists some z ∈ ∈ K {\\displaystyle \\mathbf {z} \\in K} such that p ( x + a z ) + b c < inf k ∈ ∈ K p ( x + a z + b k ) . {\\displaystyle p(x+a\\mathbf {z} )+bc~<~\\inf _{k\\in K}p(x+a\\mathbf {z} +bk).}

Adding b c {\\displaystyle bc} to both sides of the hypothesis p ( x ) + a c < inf p ( x + a K ) {\\textstyle p(x)+ac\\,<\\,\\inf _{}p(x+aK)} (where p ( x + a K ) = def { p ( x + a k ) : k ∈ ∈ K } {\\displaystyle p(x+aK)~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~\\{p(x+ak):k\\in K\\}} ) and combining that with the conclusion gives p ( x ) + a c + b c < inf p ( x + a K ) + b c ≤ ≤ p ( x + a z ) + b c < inf p ( x + a z + b K ) {\\displaystyle p(x)+ac+bc~<~\\inf _{}p(x+aK)+bc~\\leq ~p(x+a\\mathbf {z} )+bc~<~\\inf _{}p(x+a\\mathbf {z} +bK)} which yields many more inequalities, including, for instance, p ( x ) + a c + b c < p ( x + a z ) + b c < p ( x + a z + b z ) {\\displaystyle p(x)+ac+bc~<~p(x+a\\mathbf {z} )+bc~<~p(x+a\\mathbf {z} +b\\mathbf {z} )} in which an expression on one side of a strict inequality < {\\displaystyle \\,<\\,} can be obtained from the other by replacing the symbol c {\\displaystyle c} with z {\\displaystyle \\mathbf {z} } (or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged).

>>>Associated seminorm

If p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } is a real-valued sublinear function on a real vector space X {\\displaystyle X} (or if X {\\displaystyle X} is complex, then when it is considered as a real vector space) then the map q ( x ) = def max { p ( x ) , p ( − − x ) } {\\displaystyle q(x)~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~\\max\\{p(x),p(-x)\\}} defines a `F33f`_`[seminorm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Seminorm]`_`f on the real vector space X {\\displaystyle X} called the `!seminorm associated with p . {\\displaystyle p.} `!`:cite-ref-footnotenaricibeckenstein2011120-121-3-4[`F5bf`_`[3`#cite-note-footnotenaricibeckenstein2011120-121-3]`_`f] A sublinear function p {\\displaystyle p} on a real or complex vector space is a `F33f`_`[symmetric function`#symmetric-function]`_`f if and only if p = q {\\displaystyle p=q} where q ( x ) = def max { p ( x ) , p ( − − x ) } {\\displaystyle q(x)~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~\\max\\{p(x),p(-x)\\}} as before.

More generally, if p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } is a real-valued sublinear function on a (real or complex) vector space X {\\displaystyle X} then q ( x ) = def sup | u | = 1 p ( u x ) = sup { p ( u x ) : u is a unit scalar } {\\displaystyle q(x)~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~\\sup _{|u|=1}p(ux)~=~\\sup\\{p(ux):u{\\text{ is a unit scalar }}\\}} will define a `F33f`_`[seminorm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Seminorm]`_`f on X {\\displaystyle X} if this supremum is always a real number (that is, never equal to ∞ ∞ {\\displaystyle \\infty } ).

>>>Relation to linear functionals

If p {\\displaystyle p} is a sublinear function on a real vector space X {\\displaystyle X} then the following are equivalent:`:cite-ref-footnotenaricibeckenstein2011177-220-1-4[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f]

1. p {\\displaystyle p} is a `F33f`_`[linear functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_functional]`_`f.

2. for every x ∈ ∈ X , {\\displaystyle x\\in X,} p ( x ) + p ( − − x ) ≤ ≤ 0. {\\displaystyle p(x)+p(-x)\\leq 0.}

3. for every x ∈ ∈ X , {\\displaystyle x\\in X,} p ( x ) + p ( − − x ) = 0. {\\displaystyle p(x)+p(-x)=0.}

4. p {\\displaystyle p} is a minimal sublinear function.

If p {\\displaystyle p} is a sublinear function on a real vector space X {\\displaystyle X} then there exists a linear functional f {\\displaystyle f} on X {\\displaystyle X} such that f ≤ ≤ p . {\\displaystyle f\\leq p.} `:cite-ref-footnotenaricibeckenstein2011177-220-1-5[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f]

If X {\\displaystyle X} is a real vector space, f {\\displaystyle f} is a linear functional on X , {\\displaystyle X,} and p {\\displaystyle p} is a positive sublinear function on X , {\\displaystyle X,} then f ≤ ≤ p {\\displaystyle f\\leq p} on X {\\displaystyle X} if and only if f − − 1 ( 1 ) ∩ ∩ { x ∈ ∈ X : p ( x ) < 1 } = ∅ ∅ . {\\displaystyle f^{-1}(1)\\cap \\{x\\in X:p(x)<1\\}=\\varnothing .} `:cite-ref-footnotenaricibeckenstein2011177-220-1-6[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f]

>>>>Dominating a linear functional

A real-valued function f {\\displaystyle f} defined on a subset of a real or complex vector space X {\\displaystyle X} is said to be `*dominated by`* a sublinear function p {\\displaystyle p} if f ( x ) ≤ ≤ p ( x ) {\\displaystyle f(x)\\leq p(x)} for every x {\\displaystyle x} that belongs to the domain of f . {\\displaystyle f.} If f : X → → R {\\displaystyle f:X\\to \\mathbb {R} } is a real `F33f`_`[linear functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_functional]`_`f on X {\\displaystyle X} then`:cite-ref-footnoterudin199156-62-10-0[`F5bf`_`[6`#cite-note-footnoterudin199156-62-10]`_`f]`:cite-ref-footnotenaricibeckenstein2011177-220-1-7[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f] f {\\displaystyle f} is dominated by p {\\displaystyle p} (that is, f ≤ ≤ p {\\displaystyle f\\leq p} ) if and only if − − p ( − − x ) ≤ ≤ f ( x ) ≤ ≤ p ( x ) for every x ∈ ∈ X . {\\displaystyle -p(-x)\\leq f(x)\\leq p(x)\\quad {\\text{ for every }}x\\in X.} Moreover, if p {\\displaystyle p} is a seminorm or some other `*symmetric map`* (which by definition means that p ( − − x ) = p ( x ) {\\displaystyle p(-x)=p(x)} holds for all x {\\displaystyle x} ) then f ≤ ≤ p {\\displaystyle f\\leq p} if and only if | f | ≤ ≤ p . {\\displaystyle |f|\\leq p.}

`!Theorem`:cite-ref-footnotenaricibeckenstein2011177-220-1-8[`F5bf`_`[1`#cite-note-footnotenaricibeckenstein2011177-220-1]`_`f]`!—If p : X → → R {\\displaystyle p:X\\to \\mathbb {R} } be a sublinear function on a real vector space X {\\displaystyle X} and if z ∈ ∈ X {\\displaystyle z\\in X} then there exists a linear functional f {\\displaystyle f} on X {\\displaystyle X} that is dominated by p {\\displaystyle p} (that is, f ≤ ≤ p {\\displaystyle f\\leq p} ) and satisfies f ( z ) = p ( z ) . {\\displaystyle f(z)=p(z).} Moreover, if X {\\displaystyle X} is a `F33f`_`[topological vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_vector_space]`_`f and p {\\displaystyle p} is continuous at the origin then f {\\displaystyle f} is continuous.

>>>Continuity

`!Theorem`:cite-ref-footnotenaricibeckenstein2011192-193-11-0[`F5bf`_`[7`#cite-note-footnotenaricibeckenstein2011192-193-11]`_`f]`!—Suppose f : X → → R {\\displaystyle f:X\\to \\mathbb {R} } is a subadditive function (that is, f ( x + y ) ≤ ≤ f ( x ) + f ( y ) {\\displaystyle f(x+y)\\leq f(x)+f(y)} for all x , y ∈ ∈ X {\\displaystyle x,y\\in X} ). Then f {\\displaystyle f} is continuous at the origin if and only if f {\\displaystyle f} is uniformly continuous on X . {\\displaystyle X.} If f {\\displaystyle f} satisfies f ( 0 ) = 0 {\\displaystyle f(0)=0} then f {\\displaystyle f} is continuous if and only if its absolute value | f | : X → → [ 0 , ∞ ∞ ) {\\displaystyle |f|:X\\to [0,\\infty )} is continuous. If f {\\displaystyle f} is non-negative then f {\\displaystyle f} is continuous if and only if { x ∈ ∈ X : f ( x ) < 1 } {\\displaystyle \\{x\\in X:f(x)<1\\}} is open in X . {\\displaystyle X.}

Suppose X {\\displaystyle X} is a `F33f`_`[topological vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_vector_space]`_`f (TVS) over the real or complex numbers and p {\\displaystyle p} is a sublinear function on X . {\\displaystyle X.} Then the following are equivalent:`:cite-ref-footnotenaricibeckenstein2011192-193-11-1[`F5bf`_`[7`#cite-note-footnotenaricibeckenstein2011192-193-11]`_`f]

1. p {\\displaystyle p} is continuous;

2. p {\\displaystyle p} is continuous at 0;

3. p {\\displaystyle p} is uniformly continuous on X {\\displaystyle X} ;

and if p {\\displaystyle p} is positive then this list may be extended to include:

1. { x ∈ ∈ X : p ( x ) < 1 } {\\displaystyle \\{x\\in X:p(x)<1\\}} is open in X . {\\displaystyle X.}

If X {\\displaystyle X} is a real TVS, f {\\displaystyle f} is a linear functional on X , {\\displaystyle X,} and p {\\displaystyle p} is a continuous sublinear function on X , {\\displaystyle X,} then f ≤ ≤ p {\\displaystyle f\\leq p} on X {\\displaystyle X} implies that f {\\displaystyle f} is continuous.`:cite-ref-footnotenaricibeckenstein2011192-193-11-2[`F5bf`_`[7`#cite-note-footnotenaricibeckenstein2011192-193-11]`_`f]

>>>Relation to Minkowski functions and open convex sets

`!Theorem`:cite-ref-footnotenaricibeckenstein2011192-193-11-3[`F5bf`_`[7`#cite-note-footnotenaricibeckenstein2011192-193-11]`_`f]`!—If U {\\displaystyle U} is a convex open neighborhood of the origin in a `F33f`_`[topological vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_vector_space]`_`f X {\\displaystyle X} then the `F33f`_`[Minkowski functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Minkowski_functional]`_`f of U , {\\displaystyle U,} p U : X → → [ 0 , ∞ ∞ ) , {\\displaystyle p_{U}:X\\to [0,\\infty ),} is a continuous non-negative sublinear function on X {\\displaystyle X} such that U = { x ∈ ∈ X : p U ( x ) < 1 } ; {\\displaystyle U=\\left\\{x\\in X:p_{U}(x)<1\\right\\};} if in addition U {\\displaystyle U} is a `F33f`_`[balanced set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Balanced_set]`_`f then p U {\\displaystyle p_{U}} is a `F33f`_`[seminorm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Seminorm]`_`f on X . {\\displaystyle X.}

>>>>Relation to open convex sets

`!Theorem`:cite-ref-footnotenaricibeckenstein2011192-193-11-4[`F5bf`_`[7`#cite-note-footnotenaricibeckenstein2011192-193-11]`_`f]`!—Suppose that X {\\displaystyle X} is a `F33f`_`[topological vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_vector_space]`_`f (not necessarily `F33f`_`[locally convex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Locally_convex_topological_vector_space]`_`f or `F33f`_`[Hausdorff`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hausdorff_space]`_`f) over the real or complex numbers. Then the open convex subsets of X {\\displaystyle X} are exactly those that are of the form z + { x ∈ ∈ X : p ( x ) < 1 } = { x ∈ ∈ X : p ( x − − z ) < 1 } {\\displaystyle z+\\{x\\in X:p(x)<1\\}=\\{x\\in X:p(x-z)<1\\}} for some z ∈ ∈ X {\\displaystyle z\\in X} and some positive continuous sublinear function p {\\displaystyle p} on X . {\\displaystyle X.}

Proof

Let V {\\displaystyle V} be an open convex subset of X . {\\displaystyle X.} If 0 ∈ ∈ V {\\displaystyle 0\\in V} then let z := 0 {\\displaystyle z:=0} and otherwise let z ∈ ∈ V {\\displaystyle z\\in V} be arbitrary. Let p : X → → [ 0 , ∞ ∞ ) {\\displaystyle p:X\\to [0,\\infty )} be the `F33f`_`[Minkowski functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Minkowski_functional]`_`f of V − − z , {\\displaystyle V-z,} which is a continuous sublinear function on X {\\displaystyle X} since V − − z {\\displaystyle V-z} is convex, `F33f`_`[absorbing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Absorbing_set]`_`f, and open ( p {\\displaystyle p} however is not necessarily a seminorm since V {\\displaystyle V} was not assumed to be `F33f`_`[balanced`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Balanced_set]`_`f). From X = X − − z , {\\displaystyle X=X-z,} it follows that z + { x ∈ ∈ X : p ( x ) < 1 } = { x ∈ ∈ X : p ( x − − z ) < 1 } . {\\displaystyle z+\\{x\\in X:p(x)<1\\}=\\{x\\in X:p(x-z)<1\\}.} It will be shown that V = z + { x ∈ ∈ X : p ( x ) < 1 } , {\\displaystyle V=z+\\{x\\in X:p(x)<1\\},} which will complete the proof. One of the known `F33f`_`[properties of Minkowski functionals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Minkowski_functional]`_`f guarantees { x ∈ ∈ X : p ( x ) < 1 } = ( 0 , 1 ) ( V − − z ) , {\\textstyle \\{x\\in X:p(x)<1\\}=(0,1)(V-z),} where ( 0 , 1 ) ( V − − z ) = def { t x : 0 < t < 1 , x ∈ ∈ V − − z } = V − − z {\\displaystyle (0,1)(V-z)\\;{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}\\;\\{tx:0<t<1,x\\in V-z\\}=V-z} since V − − z {\\displaystyle V-z} is convex and contains the origin. Thus V − − z = { x ∈ ∈ X : p ( x ) < 1 } , {\\displaystyle V-z=\\{x\\in X:p(x)<1\\},} as desired. `F33f`_`[◼ {\displaystyle \blacksquare }`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Q.E.D.]`_`f

>>Operators

The concept can be extended to operators that are homogeneous and subadditive. This requires only that the `F33f`_`[codomain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Codomain]`_`f be, say, an `F33f`_`[ordered vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ordered_vector_space]`_`f to make sense of the conditions.

>>Computer science definition

In `F33f`_`[computer science`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Computer_science]`_`f, a function f : Z + → → R {\\displaystyle f:\\mathbb {Z} ^{+}\\to \\mathbb {R} } is called `!sublinear`! if lim n → → ∞ ∞ f ( n ) n = 0 , {\\displaystyle \\lim _{n\\to \\infty }{\\frac {f(n)}{n}}=0,} or f ( n ) ∈ ∈ o ( n ) {\\displaystyle f(n)\\in o(n)} in `F33f`_`[asymptotic notation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Big_O_notation]`_`f (notice the small o {\\displaystyle o} ). Formally, f ( n ) ∈ ∈ o ( n ) {\\displaystyle f(n)\\in o(n)} if and only if, for any given c > 0 , {\\displaystyle c>0,} there exists an N {\\displaystyle N} such that f ( n ) < c n {\\displaystyle f(n)<cn} for n ≥ ≥ N . {\\displaystyle n\\geq N.} `:cite-ref-12[`F5bf`_`[8`#cite-note-12]`_`f] That is, f {\\displaystyle f} grows slower than any linear function. The two meanings should not be confused: while a Banach functional is `F33f`_`[convex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_function]`_`f, almost the opposite is true for functions of sublinear growth: every function f ( n ) ∈ ∈ o ( n ) {\\displaystyle f(n)\\in o(n)} can be upper-bounded by a `F33f`_`[concave function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Concave_function]`_`f of sublinear growth.`:cite-ref-13[`F5bf`_`[9`#cite-note-13]`_`f]

>>See also

• `F33f`_`[Asymmetric norm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Asymmetric_norm]`_`f – Generalization of the concept of a norm

• `F33f`_`[Auxiliary normed space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Auxiliary_normed_space]`_`f

• `F33f`_`[Hahn-Banach theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hahn-Banach_theorem]`_`f – Theorem on extension of bounded linear functionalsPages displaying short descriptions of redirect targets

• `F33f`_`[Linear functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_functional]`_`f – Linear map from a vector space to its field of scalarsPages displaying short descriptions of redirect targets

• `F33f`_`[Minkowski functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Minkowski_functional]`_`f – Function made from a set

• `F33f`_`[Norm (mathematics)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Norm_(mathematics)]`_`f – Length in a vector space

• `F33f`_`[Seminorm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Seminorm]`_`f – Mathematical function

• `F33f`_`[Superadditivity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Superadditivity]`_`f – Property of a function

>>Notes

`!Proofs`!

`:cite-note-subadditivesymmetricisnonnegative-5`!proof 1.`! `F0af`_`[↑`#cite-ref-subadditivesymmetricisnonnegative-5-0]`_`f Let x ∈ ∈ X . {\\displaystyle x\\in X.} The triangle inequality and symmetry imply p ( 0 ) = p ( x + ( − − x ) ) ≤ ≤ p ( x ) + p ( − − x ) = p ( x ) + p ( x ) = 2 p ( x ) . {\\displaystyle p(0)=p(x+(-x))\\leq p(x)+p(-x)=p(x)+p(x)=2p(x).} Substituting 0 {\\displaystyle 0} for x {\\displaystyle x} and then subtracting p ( 0 ) {\\displaystyle p(0)} from both sides proves that 0 ≤ ≤ p ( 0 ) . {\\displaystyle 0\\leq p(0).} Thus 0 ≤ ≤ p ( 0 ) ≤ ≤ 2 p ( x ) {\\displaystyle 0\\leq p(0)\\leq 2p(x)} which implies 0 ≤ ≤ p ( x ) . {\\displaystyle 0\\leq p(x).} ◼ ◼ {\\displaystyle \\blacksquare }

`:cite-note-nullatzeroandsumupperbound-7`!proof 2.`! `F0af`_`[↑`#cite-ref-nullatzeroandsumupperbound-7-0]`_`f If x ∈ ∈ X {\\displaystyle x\\in X} and r := 0 {\\displaystyle r:=0} then nonnegative homogeneity implies that p ( 0 ) = p ( r x ) = r p ( x ) = 0 p ( x ) = 0. {\\displaystyle p(0)=p(rx)=rp(x)=0p(x)=0.} Consequently, 0 = p ( 0 ) = p ( x + ( − − x ) ) ≤ ≤ p ( x ) + p ( − − x ) , {\\displaystyle 0=p(0)=p(x+(-x))\\leq p(x)+p(-x),} which is only possible if 0 ≤ ≤ max { p ( x ) , p ( − − x ) } . {\\displaystyle 0\\leq \\max\\{p(x),p(-x)\\}.} ◼ ◼ {\\displaystyle \\blacksquare }

`:cite-note-reversetriangle-8`!proof 3.`! `F0af`_`[↑`#cite-ref-reversetriangle-8-0]`_`f p ( x ) = p ( y + ( x − − y ) ) ≤ ≤ p ( y ) + p ( x − − y ) , {\\displaystyle p(x)=p(y+(x-y))\\leq p(y)+p(x-y),} which happens if and only if p ( x ) − − p ( y ) ≤ ≤ p ( x − − y ) . {\\displaystyle p(x)-p(y)\\leq p(x-y).} ◼ ◼ {\\displaystyle \\blacksquare } Substituting y := − − x {\\displaystyle y:=-x} and gives p ( x ) − − p ( − − x ) ≤ ≤ p ( x − − ( − − x ) ) = p ( x + x ) ≤ ≤ p ( x ) + p ( x ) , {\\displaystyle p(x)-p(-x)\\leq p(x-(-x))=p(x+x)\\leq p(x)+p(x),} which implies − − p ( − − x ) ≤ ≤ p ( x ) {\\displaystyle -p(-x)\\leq p(x)} (positive homogeneity is not needed; the triangle inequality suffices). ◼ ◼ {\\displaystyle \\blacksquare }

`:cite-note-constantonequivclasses-9`!proof 4.`! `F0af`_`[↑`#cite-ref-constantonequivclasses-9-0]`_`f Let x ∈ ∈ X {\\displaystyle x\\in X} and k ∈ ∈ p − − 1 ( 0 ) ∩ ∩ ( − − p − − 1 ( 0 ) ) . {\\displaystyle k\\in p^{-1}(0)\\cap (-p^{-1}(0)).} It remains to show that p ( x + k ) = p ( x ) . {\\displaystyle p(x+k)=p(x).} The triangle inequality implies p ( x + k ) ≤ ≤ p ( x ) + p ( k ) = p ( x ) + 0 = p ( x ) . {\\displaystyle p(x+k)\\leq p(x)+p(k)=p(x)+0=p(x).} Since p ( − − k ) = 0 , {\\displaystyle p(-k)=0,} p ( x ) = p ( x ) − − p ( − − k ) ≤ ≤ p ( x − − ( − − k ) ) = p ( x + k ) , {\\displaystyle p(x)=p(x)-p(-k)\\leq p(x-(-k))=p(x+k),} as desired. ◼ ◼ {\\displaystyle \\blacksquare }

>>References

`:cite-note-footnotenaricibeckenstein2011177-220-1`!1.`! `F0af`_`[↑`#cite-ref-footnotenaricibeckenstein2011177-220-1-0]`_`f `F33f`_`[Narici & Beckenstein 2011`#citerefnaricibeckenstein2011]`_`f, pp. 177–220.

`:cite-note-footnoteschechter1996313-315-2`!2.`! `F0af`_`[↑`#cite-ref-footnoteschechter1996313-315-2-0]`_`f `F33f`_`[Schechter 1996`#citerefschechter1996]`_`f, pp. 313–315.

`:cite-note-footnotenaricibeckenstein2011120-121-3`!3.`! `F0af`_`[↑`#cite-ref-footnotenaricibeckenstein2011120-121-3-0]`_`f `F33f`_`[Narici & Beckenstein 2011`#citerefnaricibeckenstein2011]`_`f, pp. 120–121.

`:cite-note-footnotekubrusly2011200-4`!4.`! `F0af`_`[↑`#cite-ref-footnotekubrusly2011200-4-0]`_`f `F33f`_`[Kubrusly 2011`#citerefkubrusly2011]`_`f, p. 200.

`:cite-note-footnotenaricibeckenstein2011177-221-6`!5.`! `F0af`_`[↑`#cite-ref-footnotenaricibeckenstein2011177-221-6-0]`_`f `F33f`_`[Narici & Beckenstein 2011`#citerefnaricibeckenstein2011]`_`f, pp. 177–221.

`:cite-note-footnoterudin199156-62-10`!6.`! `F0af`_`[↑`#cite-ref-footnoterudin199156-62-10-0]`_`f `F33f`_`[Rudin 1991`#citerefrudin1991]`_`f, pp. 56–62.

`:cite-note-footnotenaricibeckenstein2011192-193-11`!7.`! `F0af`_`[↑`#cite-ref-footnotenaricibeckenstein2011192-193-11-0]`_`f `F33f`_`[Narici & Beckenstein 2011`#citerefnaricibeckenstein2011]`_`f, pp. 192–193.

`:cite-note-12`!8.`! `F0af`_`[↑`#cite-ref-12]`_`f `:citerefthomas-h-cormen-charles-e-leiserson-ronald-l-rivest-and-clifford-stein2001`a`F33f`_`[Thomas H. Cormen`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Thomas_H._Cormen]`_`f, `F33f`_`[Charles E. Leiserson`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Charles_E._Leiserson]`_`f, `F33f`_`[Ronald L. Rivest`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ronald_L._Rivest]`_`f, and `F33f`_`[Clifford Stein`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Clifford_Stein]`_`f (2001) [1990]. "3.1". `*`F33f`_`[Introduction to Algorithms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Introduction_to_Algorithms]`_`f`* (2nd ed.). MIT Press and McGraw-Hill. pp. 47–48. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-262-03293-7.`B100`F9d9{{cite book}}`f`b: CS1 maint: multiple names: authors list (link)

`:cite-note-13`!9.`! `F0af`_`[↑`#cite-ref-13]`_`f `:citerefceccherini-silbersteinsalvatorisava-huss2017`aCeccherini-Silberstein, Tullio; Salvatori, Maura; Sava-Huss, Ecaterina (2017-06-29). `*Groups, graphs, and random walks`*. Cambridge. Lemma 5.17. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9781316604403. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 948670194.`B100`F9d9{{cite book}}`f`b: CS1 maint: location missing publisher (link)

>>Bibliography

• `:citerefkubrusly2011`aKubrusly, Carlos S. (2011). `*The Elements of Operator Theory`* (Second ed.). Boston: `F33f`_`[Birkhäuser`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Birkhäuser]`_`f. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-8176-4998-2. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 710154895.

• `:citerefrudin1991`a`F33f`_`[Rudin, Walter`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Walter_Rudin]`_`f (1991). `*Functional Analysis`*. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: `F33f`_`[McGraw-Hill Science/Engineering/Math`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=McGraw-Hill_Science/Engineering/Math]`_`f. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-07-054236-5. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 21163277.

• `:citerefnaricibeckenstein2011`aNarici, Lawrence; Beckenstein, Edward (2011). `*Topological Vector Spaces`*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1584888666. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 144216834.

• `:citerefschaeferwolff1999`a`F33f`_`[Schaefer, Helmut H.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Helmut_H._Schaefer]`_`f; Wolff, Manfred P. (1999). `*Topological Vector Spaces`*. `F33f`_`[GTM`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Graduate_Texts_in_Mathematics]`_`f. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-4612-7155-0. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 840278135.

• `:citerefschechter1996`a`F33f`_`[Schechter, Eric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eric_Schechter]`_`f (1996). `*Handbook of Analysis and Its Foundations`*. San Diego, CA: Academic Press. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-12-622760-4. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 175294365.

• `:citereftr-ves2006`a`F33f`_`[Trèves, François`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=François_Trèves]`_`f (2006) [1967]. `*Topological Vector Spaces, Distributions and Kernels`*. Mineola, N.Y.: Dover Publications. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-486-45352-1. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 853623322.

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