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Infimum and supremum

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(Redirected fromLeast upper bound)
Greatest lower bound and least upper bound
A setP{\displaystyle P} of real numbers (hollow and filled circles), a subsetS{\displaystyle S} ofP{\displaystyle P} (filled circles), and the infimum ofS.{\displaystyle S.} Note that fortotally ordered finite sets, the infimum and theminimum are equal.
A setA{\displaystyle A} of real numbers (blue circles), a set of upper bounds ofA{\displaystyle A} (red diamond and circles), and the smallest such upper bound, that is, the supremum ofA{\displaystyle A} (red diamond).

In mathematics, theinfimum (abbreviatedinf;pl.:infima) of asubsetS{\displaystyle S} of apartially ordered setP{\displaystyle P} is thegreatest element inP{\displaystyle P} that is less than or equal to each element ofS,{\displaystyle S,} if such an element exists.[1] If the infimum ofS{\displaystyle S} exists, it is unique, and ifb is alower bound ofS{\displaystyle S}, thenb is less than or equal to the infimum ofS{\displaystyle S}. Consequently, the termgreatest lower bound (abbreviated asGLB) is also commonly used.[1] Thesupremum (abbreviatedsup;pl.:suprema) of a subsetS{\displaystyle S} of a partially ordered setP{\displaystyle P} is theleast element inP{\displaystyle P} that is greater than or equal to each element ofS,{\displaystyle S,} if such an element exists.[1] If the supremum ofS{\displaystyle S} exists, it is unique, and ifb is anupper bound ofS{\displaystyle S}, then the supremum ofS{\displaystyle S} is less than or equal tob. Consequently, the supremum is also referred to as theleast upper bound (orLUB).[1]

The infimum is, in a precise sense,dual to the concept of a supremum. Infima and suprema ofreal numbers are common special cases that are important inanalysis, and especially inLebesgue integration. However, the general definitions remain valid in the more abstract setting oforder theory where arbitrary partially ordered sets are considered.

The concepts of infimum and supremum are close tominimum andmaximum, but are more useful in analysis because they better characterize special sets which may haveno minimum or maximum. For instance, the set ofpositive real numbersR+{\displaystyle \mathbb {R} ^{+}} (not including0{\displaystyle 0}) does not have a minimum, because any given element ofR+{\displaystyle \mathbb {R} ^{+}} could simply be divided in half resulting in a smaller number that is still inR+.{\displaystyle \mathbb {R} ^{+}.} There is, however, exactly one infimum of the positive real numbers relative to the real numbers:0,{\displaystyle 0,} which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.

Formal definition

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supremum = least upper bound

Alower bound of a subsetS{\displaystyle S} of apartially ordered set(P,≤){\displaystyle (P,\leq )} is an elementy{\displaystyle y} ofP{\displaystyle P} such that

A lower bounda{\displaystyle a} ofS{\displaystyle S} is called aninfimum (orgreatest lower bound, ormeet) ofS{\displaystyle S} if

Similarly, anupper bound of a subsetS{\displaystyle S} of a partially ordered set(P,≤){\displaystyle (P,\leq )} is an elementz{\displaystyle z} ofP{\displaystyle P} such that

An upper boundb{\displaystyle b} ofS{\displaystyle S} is called asupremum (orleast upper bound, orjoin) ofS{\displaystyle S} if

Existence and uniqueness

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Infima and suprema do not necessarily exist. Existence of an infimum of a subsetS{\displaystyle S} ofP{\displaystyle P} can fail ifS{\displaystyle S} has no lower bound at all, or if the set of lower bounds does not contain a greatest element. (An example of this is the subset{x∈Q:x2<2}{\displaystyle \{x\in \mathbb {Q} :x^{2}<2\}} ofQ{\displaystyle \mathbb {Q} }. It has upper bounds, such as 1.5, but no supremum inQ{\displaystyle \mathbb {Q} }.)

Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, alattice is a partially ordered set in which allnonempty finite subsets have both a supremum and an infimum, and acomplete lattice is a partially ordered set in whichall subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article oncompleteness properties.

If the supremum of a subsetS{\displaystyle S} exists, it is unique. IfS{\displaystyle S} contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong toS{\displaystyle S} (or does not exist). Likewise, if the infimum exists, it is unique. IfS{\displaystyle S} contains a least element, then that element is the infimum; otherwise, the infimum does not belong toS{\displaystyle S} (or does not exist).

Relation to maximum and minimum elements

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The infimum of a subsetS{\displaystyle S} of a partially ordered setP,{\displaystyle P,} assuming it exists, does not necessarily belong toS.{\displaystyle S.} If it does, it is aminimum or least element ofS.{\displaystyle S.} Similarly, if the supremum ofS{\displaystyle S} belongs toS,{\displaystyle S,} it is amaximum or greatest element ofS.{\displaystyle S.}

For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element. For instance, for any negative real numberx,{\displaystyle x,} there is another negative real numberx2,{\displaystyle {\tfrac {x}{2}},} which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence,0{\displaystyle 0} is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.

However, the definition ofmaximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.

Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.

Minimal upper bounds

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Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not atotal one. In a totally ordered set, like the real numbers, the concepts are the same.

As an example, letS{\displaystyle S} be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets fromS{\displaystyle S} together with the set ofintegersZ{\displaystyle \mathbb {Z} } and the set of positive real numbersR+,{\displaystyle \mathbb {R} ^{+},} ordered by subset inclusion as above. Then clearly bothZ{\displaystyle \mathbb {Z} } andR+{\displaystyle \mathbb {R} ^{+}} are greater than all finite sets of natural numbers. Yet, neither isR+{\displaystyle \mathbb {R} ^{+}} smaller thanZ{\displaystyle \mathbb {Z} } nor is the converse true: both sets are minimal upper bounds but none is a supremum.

Least-upper-bound property

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Main article:Least-upper-bound property

Theleast-upper-bound property is an example of the aforementionedcompleteness properties which is typical for the set of real numbers. This property is sometimes calledDedekind completeness.

If an ordered setS{\displaystyle S} has the property that every nonempty subset ofS{\displaystyle S} having an upper bound also has a least upper bound, thenS{\displaystyle S} is said to have the least-upper-bound property. As noted above, the setR{\displaystyle \mathbb {R} } of all real numbers has the least-upper-bound property. Similarly, the setZ{\displaystyle \mathbb {Z} } of integers has the least-upper-bound property; ifS{\displaystyle S} is a nonempty subset ofZ{\displaystyle \mathbb {Z} } and there is some numbern{\displaystyle n} such that every elements{\displaystyle s} ofS{\displaystyle S} is less than or equal ton,{\displaystyle n,} then there is a least upper boundu{\displaystyle u} forS,{\displaystyle S,} an integer that is an upper bound forS{\displaystyle S} and is less than or equal to every other upper bound forS.{\displaystyle S.} Awell-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set.

An example of a set thatlacks the least-upper-bound property isQ,{\displaystyle \mathbb {Q} ,} the set of rational numbers. LetS{\displaystyle S} be the set of all rational numbersq{\displaystyle q} such thatq2<2.{\displaystyle q^{2}<2.} ThenS{\displaystyle S} has an upper bound (1000,{\displaystyle 1000,} for example, or6{\displaystyle 6}) but no least upper bound inQ{\displaystyle \mathbb {Q} }: If we supposep∈Q{\displaystyle p\in \mathbb {Q} } is the least upper bound, a contradiction is immediately deduced because between any two realsx{\displaystyle x} andy{\displaystyle y} (including2{\displaystyle {\sqrt {2}}} andp{\displaystyle p}) there exists some rationalr,{\displaystyle r,} which itself would have to be the least upper bound (ifp>2{\displaystyle p>{\sqrt {2}}}) or a member ofS{\displaystyle S} greater thanp{\displaystyle p} (ifp<2{\displaystyle p<{\sqrt {2}}}). Another example is thehyperreals; there is no least upper bound of the set of positive infinitesimals.

There is a correspondinggreatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.

If in a partially ordered setP{\displaystyle P} every bounded subset has a supremum, this applies also, for any setX,{\displaystyle X,} in the function space containing all functions fromX{\displaystyle X} toP,{\displaystyle P,} wheref≤g{\displaystyle f\leq g} if and only iff(x)≤g(x){\displaystyle f(x)\leq g(x)} for allx∈X.{\displaystyle x\in X.} For example, it applies for real functions, and, since these can be considered special cases of functions, for realn{\displaystyle n}-tuples and sequences of real numbers.

Theleast-upper-bound property is an indicator of the suprema.

Infima and suprema of real numbers

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Inanalysis, infima and suprema of subsetsS{\displaystyle S} of thereal numbers are particularly important. For instance, the negativereal numbers do not have a greatest element, and their supremum is0{\displaystyle 0} (which is not a negative real number).[1]Thecompleteness of the real numbers implies (and is equivalent to) that any bounded nonempty subsetS{\displaystyle S} of the real numbers has an infimum and a supremum. IfS{\displaystyle S} is not bounded below, one often formally writesinfS=−∞.{\displaystyle \inf _{}S=-\infty .} IfS{\displaystyle S} isempty, one writesinfS=+∞.{\displaystyle \inf _{}S=+\infty .}

Properties

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IfA{\displaystyle A} is any set of real numbers thenA≠∅{\displaystyle A\neq \varnothing } if and only ifsupA≥infA,{\displaystyle \sup A\geq \inf A,} and otherwise−∞=sup∅<inf∅=∞.{\displaystyle -\infty =\sup \varnothing <\inf \varnothing =\infty .}[2]

Set inclusion

IfA⊆B{\displaystyle A\subseteq B} are sets of real numbers theninfA≥infB{\displaystyle \inf A\geq \inf B} (ifA=∅{\displaystyle A=\varnothing } this reads asinfB≤∞{\displaystyle \inf B\leq \infty }) andsupA≤supB.{\displaystyle \sup A\leq \sup B.}

Image under functionsIff:R→R{\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is a nonincreasing function, thenf(inf(S))≤inf(f(S)){\displaystyle f(\inf(S))\leq \inf(f(S))} andsup(f(S)){\displaystyle \sup(f(S))}, where the image is defined asf(S)=def{f(s):s∈S}.{\displaystyle f(S)\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\{f(s):s\in S\}.}

Identifying infima and suprema

If the infimum ofA{\displaystyle A} exists (that is,infA{\displaystyle \inf A} is a real number) and ifp{\displaystyle p} is any real number thenp=infA{\displaystyle p=\inf A} if and only ifp{\displaystyle p} is a lower bound and for everyϵ>0{\displaystyle \epsilon >0} there is anaϵ∈A{\displaystyle a_{\epsilon }\in A} withaϵ<p+ϵ.{\displaystyle a_{\epsilon }<p+\epsilon .} Similarly, ifsupA{\displaystyle \sup A} is a real number and ifp{\displaystyle p} is any real number thenp=supA{\displaystyle p=\sup A} if and only ifp{\displaystyle p} is an upper bound and if for everyϵ>0{\displaystyle \epsilon >0} there is anaϵ∈A{\displaystyle a_{\epsilon }\in A} withaϵ>p−ϵ.{\displaystyle a_{\epsilon }>p-\epsilon .}

Relation to limits of sequences

IfS≠∅{\displaystyle S\neq \varnothing } is any non-empty set of real numbers then there always exists a non-decreasing sequences1≤s2≤⋯{\displaystyle s_{1}\leq s_{2}\leq \cdots } inS{\displaystyle S} such thatlimn→∞sn=supS.{\displaystyle \lim _{n\to \infty }s_{n}=\sup S.} Similarly, there will exist a (possibly different) non-increasing sequences1≥s2≥⋯{\displaystyle s_{1}\geq s_{2}\geq \cdots } inS{\displaystyle S} such thatlimn→∞sn=infS.{\displaystyle \lim _{n\to \infty }s_{n}=\inf S.}In particular, the infimum and supremum of a set belong to itsclosure ifinfS∈R{\displaystyle \inf S\in \mathbb {R} } theninfS∈S¯{\displaystyle \inf S\in {\bar {S}}} and ifsupS∈R{\displaystyle \sup S\in \mathbb {R} } thensupS∈S¯{\displaystyle \sup S\in {\bar {S}}}

Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact fromtopology that iff{\displaystyle f} is acontinuous function ands1,s2,…{\displaystyle s_{1},s_{2},\ldots } is a sequence of points in its domain that converges to a pointp,{\displaystyle p,} thenf(s1),f(s2),…{\displaystyle f\left(s_{1}\right),f\left(s_{2}\right),\ldots } necessarily converges tof(p).{\displaystyle f(p).} It implies that iflimn→∞sn=supS{\displaystyle \lim _{n\to \infty }s_{n}=\sup S} is a real number (where alls1,s2,…{\displaystyle s_{1},s_{2},\ldots } are inS{\displaystyle S}) and iff{\displaystyle f} is a continuous function whose domain containsS{\displaystyle S} andsupS,{\displaystyle \sup S,} thenf(supS)=f(limn→∞sn)=limn→∞f(sn),{\displaystyle f(\sup S)=f\left(\lim _{n\to \infty }s_{n}\right)=\lim _{n\to \infty }f\left(s_{n}\right),}which (for instance) guarantees[note 1] thatf(supS){\displaystyle f(\sup S)} is anadherent point of the setf(S)=def{f(s):s∈S}.{\displaystyle f(S)\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\{f(s):s\in S\}.}If in addition to what has been assumed, the continuous functionf{\displaystyle f} is also an increasing ornon-decreasing function, then it is even possible to conclude thatsupf(S)=f(supS).{\displaystyle \sup f(S)=f(\sup S).} This may be applied, for instance, to conclude that wheneverg{\displaystyle g} is a real (orcomplex) valued function with domainΩ≠∅{\displaystyle \Omega \neq \varnothing } whosesup norm‖g‖∞=defsupx∈Ω|g(x)|{\displaystyle \|g\|_{\infty }\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\sup _{x\in \Omega }|g(x)|} is finite, then for every non-negative real numberq,{\displaystyle q,}‖g‖∞q =def (supx∈Ω|g(x)|)q=supx∈Ω(|g(x)|q){\displaystyle \|g\|_{\infty }^{q}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\sup _{x\in \Omega }|g(x)|\right)^{q}=\sup _{x\in \Omega }\left(|g(x)|^{q}\right)}since the mapf:[0,∞)→R{\displaystyle f:[0,\infty )\to \mathbb {R} } defined byf(x)=xq{\displaystyle f(x)=x^{q}} is a continuous non-decreasing function whose domain[0,∞){\displaystyle [0,\infty )} always containsS:={|g(x)|:x∈Ω}{\displaystyle S:=\{|g(x)|:x\in \Omega \}} andsupS=def‖g‖∞.{\displaystyle \sup S\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\|g\|_{\infty }.}

Although this discussion focused onsup,{\displaystyle \sup ,} similar conclusions can be reached forinf{\displaystyle \inf } with appropriate changes (such as requiring thatf{\displaystyle f} be non-increasing rather than non-decreasing). Othernorms defined in terms ofsup{\displaystyle \sup } orinf{\displaystyle \inf } include theweakLp,w{\displaystyle L^{p,w}} space norms (for1≤p<∞{\displaystyle 1\leq p<\infty }), the norm onLebesgue spaceL∞(Ω,μ),{\displaystyle L^{\infty }(\Omega ,\mu ),} andoperator norms. Monotone sequences inS{\displaystyle S} that converge tosupS{\displaystyle \sup S} (or toinfS{\displaystyle \inf S}) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.

Arithmetic operations on sets

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The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets. Throughout,A,B⊆R{\displaystyle A,B\subseteq \mathbb {R} } are sets of real numbers.

Sum of sets

TheMinkowski sum of two setsA{\displaystyle A} andB{\displaystyle B} of real numbers is the setA+B := {a+b:a∈A,b∈B}{\displaystyle A+B~:=~\{a+b:a\in A,b\in B\}} consisting of all possible arithmetic sums of pairs of numbers, one from each set. The infimum and supremum of the Minkowski sum satisfy, ifA≠∅≠B{\displaystyle A\neq \varnothing \neq B}inf(A+B)=(infA)+(infB){\displaystyle \inf(A+B)=(\inf A)+(\inf B)} andsup(A+B)=(supA)+(supB).{\displaystyle \sup(A+B)=(\sup A)+(\sup B).}

Product of sets

The multiplication of two setsA{\displaystyle A} andB{\displaystyle B} of real numbers is defined similarly to their Minkowski sum:A⋅B := {a⋅b:a∈A,b∈B}.{\displaystyle A\cdot B~:=~\{a\cdot b:a\in A,b\in B\}.}

IfA{\displaystyle A} andB{\displaystyle B} are nonempty sets of positive real numbers theninf(A⋅B)=(infA)⋅(infB){\displaystyle \inf(A\cdot B)=(\inf A)\cdot (\inf B)} and similarly for supremasup(A⋅B)=(supA)⋅(supB).{\displaystyle \sup(A\cdot B)=(\sup A)\cdot (\sup B).}[3]

Scalar product of a set

The product of a real numberr{\displaystyle r} and a setB{\displaystyle B} of real numbers is the setrB := {r⋅b:b∈B}.{\displaystyle rB~:=~\{r\cdot b:b\in B\}.}

Ifr>0{\displaystyle r>0} theninf(r⋅A)=r(infA) and sup(r⋅A)=r(supA),{\displaystyle \inf(r\cdot A)=r(\inf A)\quad {\text{ and }}\quad \sup(r\cdot A)=r(\sup A),}while ifr<0{\displaystyle r<0} theninf(r⋅A)=r(supA) and sup(r⋅A)=r(infA).{\displaystyle \inf(r\cdot A)=r(\sup A)\quad {\text{ and }}\quad \sup(r\cdot A)=r(\inf A).}In the caser=0{\displaystyle r=0},one has, ifA≠∅{\displaystyle A\neq \varnothing }inf(0⋅A)=0 and sup(0⋅A)=0{\displaystyle \inf(0\cdot A)=0\quad {\text{ and }}\quad \sup(0\cdot A)=0} Usingr=−1{\displaystyle r=-1} and the notation−A:=(−1)A={−a:a∈A},{\textstyle -A:=(-1)A=\{-a:a\in A\},} it follows that,inf(−A)=−supA and sup(−A)=−infA.{\displaystyle \inf(-A)=-\sup A\quad {\text{ and }}\quad \sup(-A)=-\inf A.}

Multiplicative inverse of a set

For any setS{\displaystyle S} that does not contain0,{\displaystyle 0,} let1S :={1s:s∈S}.{\displaystyle {\frac {1}{S}}~:=\;\left\{{\tfrac {1}{s}}:s\in S\right\}.}

IfS⊆(0,∞){\displaystyle S\subseteq (0,\infty )} is non-empty then1supS = inf1S{\displaystyle {\frac {1}{\sup _{}S}}~=~\inf _{}{\frac {1}{S}}} where this equation also holds whensupS=∞{\displaystyle \sup _{}S=\infty } if the definition1∞:=0{\displaystyle {\frac {1}{\infty }}:=0} is used.[note 2] This equality may alternatively be written as1sups∈Ss=infs∈S1s.{\displaystyle {\frac {1}{\displaystyle \sup _{s\in S}s}}=\inf _{s\in S}{\tfrac {1}{s}}.} Moreover,infS=0{\displaystyle \inf _{}S=0} if and only ifsup1S=∞,{\displaystyle \sup _{}{\tfrac {1}{S}}=\infty ,} where if[note 2]infS>0,{\displaystyle \inf _{}S>0,} then1infS=sup1S.{\displaystyle {\tfrac {1}{\inf _{}S}}=\sup _{}{\tfrac {1}{S}}.}

Duality

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If one denotes byPop{\displaystyle P^{\operatorname {op} }} the partially-ordered setP{\displaystyle P} with theopposite order relation; that is, for allx and y,{\displaystyle x{\text{ and }}y,} declare:x≤y in Pop if and only if x≥y in P,{\displaystyle x\leq y{\text{ in }}P^{\operatorname {op} }\quad {\text{ if and only if }}\quad x\geq y{\text{ in }}P,}then infimum of a subsetS{\displaystyle S} inP{\displaystyle P} equals the supremum ofS{\displaystyle S} inPop{\displaystyle P^{\operatorname {op} }} and vice versa.

For subsets of the real numbers, another kind of duality holds:infS=−sup(−S),{\displaystyle \inf S=-\sup(-S),} where−S:={−s : s∈S}.{\displaystyle -S:=\{-s~:~s\in S\}.}

Examples

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Infima

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Suprema

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In the last example, the supremum of a set ofrationals isirrational, which means that the rationals areincomplete.

One basic property of the supremum issup{f(t)+g(t):t∈A} ≤ sup{f(t):t∈A}+sup{g(t):t∈A}{\displaystyle \sup\{f(t)+g(t):t\in A\}~\leq ~\sup\{f(t):t\in A\}+\sup\{g(t):t\in A\}}for anyfunctionalsf{\displaystyle f} andg.{\displaystyle g.}

The supremum of a subsetS{\displaystyle S} of(N,∣){\displaystyle (\mathbb {N} ,\mid \,)} where∣{\displaystyle \,\mid \,} denotes "divides", is thelowest common multiple of the elements ofS.{\displaystyle S.}

The supremum of a setS{\displaystyle S} containing subsets of some setX{\displaystyle X} is theunion of the subsets when considering the partially ordered set(P(X),⊆){\displaystyle (P(X),\subseteq )}, whereP{\displaystyle P} is thepower set ofX{\displaystyle X} and⊆{\displaystyle \,\subseteq \,} issubset.

See also

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Wikimedia Commons has media related toInfimum and supremum.

Notes

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  1. ^Sincef(s1),f(s2),…{\displaystyle f\left(s_{1}\right),f\left(s_{2}\right),\ldots } is a sequence inf(S){\displaystyle f(S)} that converges tof(supS),{\displaystyle f(\sup S),} this guarantees thatf(supS){\displaystyle f(\sup S)} belongs to theclosure off(S).{\displaystyle f(S).}
  2. ^abThe definition1∞:=0{\displaystyle {\tfrac {1}{\infty }}:=0} is commonly used with theextended real numbers; in fact, with this definition the equality1supS=inf1S{\displaystyle {\tfrac {1}{\sup _{}S}}=\inf _{}{\tfrac {1}{S}}} will also hold for any non-empty subsetS⊆(0,∞].{\displaystyle S\subseteq (0,\infty ].} However, the notation10{\displaystyle {\tfrac {1}{0}}} is usually left undefined, which is why the equality1infS=sup1S{\displaystyle {\tfrac {1}{\inf _{}S}}=\sup _{}{\tfrac {1}{S}}} is given only for wheninfS>0.{\displaystyle \inf _{}S>0.}

References

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  1. ^abcdeRudin, Walter (1976). "Chapter 1 The Real and Complex Number Systems".Principles of Mathematical Analysis(print) (3rd ed.). McGraw-Hill. p. 4.ISBN 0-07-054235-X.
  2. ^Rockafellar & Wets 2009, pp. 1–2.
  3. ^Zakon, Elias (2004).Mathematical Analysis I. Trillia Group. pp. 39–42.

External links

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