Convex cone

A subset $C$ of a vector space is a cone if $x\backslash in\; C$ implies $sx\backslash in\; C$ for every $s>0$. Here $s>0$ refers to (strict) positivity in the scalar field.

Some other authors require $[0,\backslash infty)C\backslash subset\; C$ or even $0\backslash in\; C$.

Some require a cone to be convex and/or satisfy $C\backslash cap-C\backslash subset\backslash \{0\backslash \}$.

The conical hull of a set $C$ is defined as the smallest convex cone that contains $C\backslash cup\backslash \{0\backslash \}$. Therefore, it need not be the smallest cone that contains $C\backslash cup\backslash \{0\backslash \}$.

Wedge may refer to what we call cones (when "cone" is reserved for something stronger), or just to a subset of them, depending on the author.

A subset $C$ of a vector space $V$ over an ordered field $F$ is a cone (or sometimes called a linear cone) if for each $x$ in $C$ and positive scalar $\backslash alpha$ in $F$, the product $\backslash alpha\; x$ is in $C$.{{Cite book|url=https://books.google.com/books?id=x7isojLkDTcC|title=Matrix Mathematics: Theory, Facts, and Formulas|last=Bernstein|first=Dennis S.|date=2009-07-26|publisher=Princeton University Press|isbn=978-0691140391|pages=97|language=en|edition=Second}} Note that some authors define cone with the scalar $\backslash alpha$ ranging over all non-negative scalars (rather than all positive scalars, which does not include 0).{{cite book|author=C. Zalinescu|title=Convex Analysis in General Vector Spaces|url=https://books.google.com/books?id=a91q1q1djMwC|date=1 January 2002|publisher=World Scientific|isbn=978-981-238-067-8|page=1}} Some authors even require $0\backslash in\; C$, thus excluding the empty set.{{Cite book |author=W. Fenchel |title=CONVEX CONES, SETS, AND FUNCTIONS |date=September 1953 |publisher=Princeton University, Department of Mathematics|page=1}}

Therefore, $[0,\backslash infty)$ is a cone, $\backslash varnothing$ is a cone only according to the 1st and 2nd definition above, and $(0,\backslash infty)$ is a cone only according to the 1st definition above. All of them are convex (see below).

A cone $C$ is a convex cone if $\backslash alpha\; x+\backslash beta\; y$ belongs to $C$, for any positive scalars $\backslash alpha$, $\backslash beta$, and any $x$, $y$ in $C$.{{Cite book|url=https://books.google.com/books?id=cX-TGJb1gfkC|title=Linear Algebra|last=Nef|first=Walter|date=1988-01-01|publisher=Courier Corporation| isbn=9780486657721 |pages=35|language=en}}{{Cite book|url=https://books.google.com/books?id=WHjO9K6xEm4C|title=Encyclopedic Dictionary of Mathematics| last=Itô|first=Kiyosi|date=1993-01-01|publisher=MIT Press|isbn=9780262590204|language=en}}

A cone $C$ is convex if and only if $C+C\backslash subseteq\; C$.

This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real numbers. Also note that the scalars in the definition are positive meaning that the origin does not have to belong to $C$. Some authors use a definition that ensures the origin belongs to $C$.{{Cite book|url=https://books.google.com/books?id=jzpzBwAAQBAJ|title=Convex Analysis|last=Rockafellar|first=Ralph Tyrell|date=2015-04-29|publisher=Princeton University Press|isbn=9781400873173|pages=13|language=en}} Because of the scaling parameters $\backslash alpha$ and $\backslash beta$, cones are infinite in extent and not bounded.

If $C$ is a convex cone, then for any positive scalar $\backslash alpha$ and any $x$ in $C$ the vector $\backslash alpha\; x\; =\; \backslash tfrac\{\backslash alpha\}\{2\}x\; +\; \backslash tfrac\{\backslash alpha\}\{2\}x\; \backslash in\; C.$ So a convex cone is a special case of a linear cone as defined above.

It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under addition. More succinctly, a set $C$ is a convex cone if and only if $\backslash alpha\; C=C$ for every positive scalar $\backslash alpha$ and $C+C=C$.

A face of a convex cone $C$ is a subset $F$ of $C$ such that $F$ is also a convex cone, and for any vectors $x,y$ in $C$ with $x+y$ in $F$, $x$ and $y$ must both be in $F$.{{sfn | Rockafellar| 1997 | p=162}} For example, $C$ itself is a face of $C$. The origin $\backslash \{\; 0\backslash \}$ is a face of $C$ if $C$ contains no line (so $C$ is "strictly convex", or "salient", as defined below). The origin and $C$ are sometimes called the trivial faces of $C$. A ray (the set of nonnegative multiples of a nonzero vector) is called an extremal ray if it is a face of $C$.

Let $C$ be a closed, strictly convex cone in $\backslash R^n$. Suppose that $C$ is more than just the origin. Then $C$ is the convex hull of its extremal rays.{{sfn | Rockafellar| 1997 | p=166}}

File:Circular-pyramid.png

File:Polyhedral-cone.png

File:Cone-not-convex.png

• For a vector space $V$, every linear subspace of $V$ is a convex cone. In particular, the space $V$ itself and the origin $\backslash \{\; 0\backslash \}$ are convex cones in $V$. For authors who do not require a convex cone to contain the origin, the empty set $\backslash emptyset$ is also a convex cone.
• The conical hull of a finite or infinite set of vectors in $\backslash R^n$ is a convex cone.
• The tangent cones of a convex set are convex cones.
• The set $$\backslash left\; \backslash \{\; x\; \backslash in\; \backslash R^2\; \backslash mid\; x\_2\; \backslash geq\; 0,\; x\_1\; =\; 0\; \backslash right\; \backslash \}\; \backslash cup\; \backslash left\; \backslash \{\; x\; \backslash in\; \backslash R^2\; \backslash mid\; x\_1\; \backslash geq\; 0,\; x\_2\; =\; 0\; \backslash right\; \backslash \}$$ is a cone but not a convex cone.
• The norm cone $$C\; =\; \backslash left\; \backslash \{\; (x,\; r)\; \backslash in\; \backslash R^\{d+1\}\; \backslash mid\; \backslash |x\backslash |\; \backslash leq\; r\; \backslash right\; \backslash \}$$ is a convex cone. (For $d=2$, this is the round cone in the figure.) Each extremal ray of $C$ is spanned by a vector $(x,1)$ with $\backslash |x\backslash |=1$ (so $x$ is a point in the sphere $S^\{d-1\}$). These rays are in fact the only nontrivial faces of $C$.
• The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one.
• The class of convex cones is also closed under arbitrary linear maps. In particular, if $C$ is a convex cone, so is its opposite $-C$, and $C\; \backslash cap\; -C$ is the largest linear subspace contained in $C$.
• The set of positive semidefinite matrices.
• The set of nonnegative continuous functions is a convex cone.

An affine convex cone is the set resulting from applying an affine transformation to a convex cone.{{Cite book|url=https://books.google.com/books?id=hIYKBwAAQBAJ|title=Fundamentals of Convex Analysis|last1=Hiriart-Urruty|first1=Jean-Baptiste|last2=Lemaréchal|first2=Claude|date=2012-12-06|publisher=Springer Science & Business Media|isbn=9783642564680|language=en}} A common example is translating a convex cone by a point {{nowrap|p: p + C}}. Technically, such transformations can produce non-cones. For example, unless {{nowrap|p {{=}} 0}}, {{nowrap|p + C}} is not a linear cone. However, it is still called an affine convex cone.

A (linear) hyperplane is a set in the form $\backslash \{\; x\; \backslash in\; V\; \backslash mid\; f(x)\; =\; c\backslash \}$ where f is a linear functional on the vector space V. A closed half-space is a set in the form $\backslash \{\; x\; \backslash in\; V\; \backslash mid\; f(x)\; \backslash leq\; c\backslash \}$ or $\backslash \{\; x\; \backslash in\; V\; \backslash mid\; f(x)\; \backslash geq\; c\backslash \},$ and likewise an open half-space uses strict inequality.{{Cite book|url=https://books.google.com/books?id=4hIq6ExH7NoC|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|last1=Aliprantis|first1=Charalambos D.|last2=Border|first2=Kim C.|date=2007-05-02|publisher=Springer Science & Business Media| isbn=9783540326960| pages =197 |language=en}}{{Cite book|url=https://books.google.com/books?id=jzpzBwAAQBAJ|title=Convex Analysis|last=Rockafellar|first=Ralph Tyrell| date=2015-04-29|publisher=Princeton University Press|isbn=9781400873173|pages=10|language=en}}

Half-spaces (open or closed) are affine convex cones. Moreover (in finite dimensions), any convex cone C that is not the whole space V must be contained in some closed half-space H of V; this is a special case of Farkas' lemma.

Polyhedral cones are special kinds of cones that can be defined in several ways:{{Cite Lovasz Plummer}}{{Rp|256–257}}

• A cone $C$ is polyhedral if it is the conical hull of finitely many vectors (this property is also called finitely-generated).{{Cite book|last1=Loera|first1=Jesús A. De|url=https://books.google.com/books?id=eVHg7aECS7UC|title=Algebraic and Geometric Ideas in the Theory of Discrete Optimization|last2=Hemmecke|first2=Raymond|last3=Köppe|first3=Matthias|date=2012-01-01|publisher=SIAM|isbn=9781611972443|language=en}}{{Cite book|last=Schrijver|first=Alexander|url=https://books.google.com/books?id=zEzW5mhppB8C|title=Theory of Linear and Integer Programming|date=1998-07-07|publisher=John Wiley & Sons|isbn=9780471982326|language=en}} I.e., there is a set of vectors $\backslash \{v\_1,\; \backslash ldots,\; v\_k\backslash \}\backslash subset\; \backslash R^n$ so that $C\; =\; \backslash \{\; a\_1\; v\_1\; +\; \backslash cdots\; +\; a\_kv\_k\; \backslash mid\; a\_i\; \backslash in\; \backslash R\_\{\backslash ge\; 0\}\backslash \}$.
• A cone is polyhedral if it is the intersection of a finite number of half-spaces which have 0 on their boundary (the equivalence between these first two definitions was proved by Weyl in 1935).{{cite journal

| last = Weyl | first = H.

| doi = 10.1007/BF01292722

| journal = Commentarii Mathematici Helvetici

| pages = 290–306

| title = Elementare Theorie der konvexen Polyeder

| volume = 7

| year = 1935}}{{cite journal

| last = Mirkil | first = H.

| doi = 10.4153/CJM-1957-001-5

| journal = Canadian Journal of Mathematics

| mr = 83761

| pages = 1–4

| title = New characterizations of polyhedral cones

| volume = 9

| year = 1957}}

• A cone $C$ is polyhedral if there is some matrix $A\backslash in\; \backslash R^\{m\backslash times\; n\}$ such that $C\; =\; \backslash \{x\; \backslash in\; \backslash R^n\; \backslash mid\; Ax\; \backslash geq\; 0\; \backslash \}$.
• A cone is polyhedral if it is the solution set of a system of homogeneous linear inequalities. Algebraically, each inequality is defined by a row of the matrix $A$. Geometrically, each inequality defines a halfspace that passes through the origin.

Every finitely generated cone is a polyhedral cone, and every polyhedral cone is a finitely generated cone. Every polyhedral cone has a unique representation as a conical hull of its extremal generators, and a unique representation of intersections of halfspaces, given each linear form associated with the halfspaces also define a support hyperplane of a facet.{{cite book|last1=Bruns|first1=Winfried|last2=Gubeladze|first2=Joseph|title=Polytopes, Rings and K-Theory|url=https://archive.org/details/polytopesringskt00albr|url-access=limited|date=2009|publisher=Springer Monographs in Mathematics|isbn=9780387763552|page=[https://archive.org/details/polytopesringskt00albr/page/n14 3]|edition=1}}

Each face of a polyhedral cone is spanned by some subset of its extremal generators. As a result, a polyhedral cone has only finitely many faces.

Polyhedral cones play a central role in the representation theory of polyhedra. For instance, the decomposition theorem for polyhedra states that every polyhedron can be written as the Minkowski sum of a convex polytope and a polyhedral cone.{{Cite book|url=https://books.google.com/books?id=zEzW5mhppB8C|title=Theory of Linear and Integer Programming| last=Schrijver |first=Alexander |date=1998-07-07|publisher=John Wiley & Sons|isbn=9780471982326|pages=88–89|language=en}}{{Cite book|url=https://books.google.com/books?id=antqBQAAQBAJ| title=Integer Programming|last1=Conforti|first1=Michele|last2=Cornuejols|first2=Gerard|last3=Zambelli|first3=Giacomo|date=2014-11-15| publisher=Springer |isbn=9783319110080|pages=111|language=en}} Polyhedral cones also play an important part in proving the related Finite Basis Theorem for polytopes which shows that every polytope is a polyhedron and every bounded polyhedron is a polytope.{{Cite book|url=https://books.google.com/books?id=X-klBQAAQBAJ&q=finite+basis+theorem+polyhedra |title=Combinatorial Optimization: Theory and Algorithms| last1=Korte| first1=Bernhard| last2=Vygen| first2=Jens|date=2013-11-11|publisher=Springer Science & Business Media|isbn=9783662217115|pages=61|language=en}}{{Cite book| url=https://books.google.com/books?id=t3oZBwAAQBAJ|title=Monomial Algebras, Second Edition|last=Villarreal|first=Rafael|date=2015-03-26|publisher=CRC Press| isbn=9781482234701|pages=9|language=en}}

The two representations of a polyhedral cone - by inequalities and by vectors - may have very different sizes. For example, consider the cone of all non-negative $n$-by-$n$ matrices with equal row and column sums. The inequality representation requires $n^2$ inequalities and $2n-1$ equations, but the vector representation requires $n!$ vectors (see the Birkhoff-von Neumann Theorem). The opposite can also happen - the number of vectors may be polynomial while the number of inequalities is exponential.{{Rp|256}}

The two representations together provide an efficient way to decide whether a given vector is in the cone: to show that it is in the cone, it is sufficient to present it as a conic combination of the defining vectors; to show that it is not in the cone, it is sufficient to present a single defining inequality that it violates. This fact is known as Farkas' lemma.

A subtle point in the representation by vectors is that the number of vectors may be exponential in the dimension, so the proof that a vector is in the cone might be exponentially long. Fortunately, Carathéodory's theorem guarantees that every vector in the cone can be represented by at most $d$ defining vectors, where $d$ is the dimension of the space.

According to the above definition, if C is a convex cone, then C ∪ {0} is a convex cone, too. A convex cone is said to be {{visible anchor|pointed}} if 0 is in C, and {{visible anchor|blunt}} if 0 is not in C.{{Cite book|url=https://books.google.com/books?id=eP9yuZMaJjQC| title= Optimality Conditions in Convex Optimization: A Finite-Dimensional View| last1=Dhara|first1=Anulekha|last2=Dutta|first2=Joydeep|date=2011-10-17| publisher =CRC Press|isbn=9781439868225|pages=243|language=en}} Some authors use "pointed" for $C\backslash cap-C=\backslash \{0\backslash \}$ or salient (see below).https://healy.econ.ohio-state.edu/kcb/Ec181/Lecture03.pdf

Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β.

A cone is called flat if it contains some nonzero vector x and its opposite −x, meaning C contains a linear subspace of dimension at least one, and salient (or strictly convex) otherwise.{{Cite book|url=https://books.google.com/books?id=go59BgAAQBAJ|title=Optimization: A Theory of Necessary Conditions| last=Neustadt| first=Lucien W.|date=2015-03-08|publisher=Princeton University Press|isbn=9781400870530|pages=6|language=en}}{{Cite book| url=https://books.google.com/books?id=fdhi90F0HvcC|title=Functional Analysis: Theory and Applications|last=Edwards|first=R. E.|date=2012-10-25|publisher=Courier Corporation|isbn=9780486145105|pages=135|language=en}}

A blunt convex cone is necessarily salient, but the converse is not necessarily true.

A convex cone C is salient if and only if C ∩ −C ⊆ {0}.

A cone C is said to be generating if $C-C=\backslash \{x-y\backslash mid\; x\backslash in\; C,\; y\backslash in\; C\backslash \}$ equals the whole vector space.{{sfn | Schaefer|Wolff| 1999 | pp=205–209}}

Some authors require salient cones to be pointed.{{Cite book|url=https://books.google.com/books?id=y_IJuH1bs8YC|title=Generalized Convexity and Generalized Monotonicity: Proceedings of the 6th International Symposium on Generalized Convexity/Monotonicity, Samos, September 1999| last1=Hadjisavvas| first1=Nicolas|last2=Martinez-Legaz |first2=Juan E.|last3=Penot|first3=Jean-Paul|date=2001-04-10|publisher=Springer Science & Business Media| isbn= 9783540418061| pages=238|language=en}}

The term "pointed" is also often used to refer to a closed cone that contains no complete line (i.e., no nontrivial subspace of the ambient vector space V, or what is called a salient cone).{{Cite book|url=https://books.google.com/books?id=cxL3jL7ONjQC|title=Convex Analysis and Monotone Operator Theory in Hilbert Spaces|last1=Bauschke|first1=Heinz H.|last2=Combettes|first2=Patrick L.|date=2011-04-19| publisher=Springer Science & Business Media|isbn=9781441994677|pages=88|language=en}}{{Cite book|url=https://books.google.com/books?id=fGg7AAAAIAAJ|title=Introduction to Linear and Convex Programming|last=Cameron|first=Neil|date=1985-09-05|publisher=CUP Archive| isbn=9780521312073| pages=32| language=en}}{{Cite book|url=https://books.google.com/books?id=7s_gBwAAQBAJ|title=Linear Programming: Mathematics, Theory and Algorithms| last=Panik|first=M. J.|date=2013-12-01|publisher=Springer Science & Business Media|isbn=9781461334347|pages=40|language=en}}

The term proper (convex) cone is variously defined, depending on the context and author. It often means a cone that satisfies other properties like being convex, closed, pointed, salient, and full-dimensional.{{Cite book|url=https://books.google.com/books?id=VxuRF4NhjqkC|title=Convex Optimization & Euclidean Distance Geometry|last=Dattorro|first=Jon|date=2005-01-01|publisher=Meboo Publishing USA|isbn=9780976401308|pages=96|language=en}}{{Cite book| url=https://books.google.com/books?id=Zm3mCAAAQBAJ|title=Mainstream Mathematical Economics in the 20th Century|last=Nicola|first=PierCarlo|date=2013-03-14| publisher=Springer Science & Business Media|isbn=9783662042380|pages=125|language=en}}{{Cite book|url=https://books.google.com/books?id=f966BQAAQBAJ| title=Harmonic Analysis on Exponential Solvable Lie Groups|last1=Fujiwara|first1=Hidenori|last2=Ludwig|first2=Jean|date=2014-12-05| publisher =Springer |isbn=9784431552888|pages=246|language=en}} Because of these varying definitions, the context or source should be consulted for the definition of these terms.

A type of cone of particular interest to pure mathematicians is the partially ordered set of rational cones. "Rational cones are important objects in toric algebraic geometry, combinatorial commutative algebra, geometric combinatorics, integer programming.".{{cite journal|last1=Gubeladze|first1=Joseph|last2=Michałek|first2=Mateusz|title=The poset of rational cones|journal=Pacific Journal of Mathematics|date=1 January 2018|volume=292|issue=1|pages=103–115|doi=10.2140/pjm.2018.292.103|arxiv=1606.02083|s2cid=119639952}} This object arises when we study cones in $\backslash R^d$ together with the lattice $\backslash Z^d$. A cone is called rational (here we assume "pointed", as defined above) whenever its generators all have integer coordinates, i.e., if $C$ is a rational cone, then $C\; =\; \backslash \{\; a\_1\; v\_1\; +\; \backslash cdots\; +\; a\_k\; v\_k\; \backslash mid\; a\_i\; \backslash in\; \backslash R\_+\; \backslash \}$ for some $v\_i\; \backslash in\; \backslash Z^d$.

{{Main article|dual cone}}

Let C ⊂ V be a set, not necessarily a convex set, in a real vector space V equipped with an inner product. The (continuous or topological) dual cone to C is the set

:$C^*\; =\; \backslash \{\; v\backslash in\; V\; \backslash mid\; \backslash forall\; w\backslash in\; C,\; \backslash langle\; w,v\; \backslash rangle\; \backslash ge\; 0\; \backslash \},$

which is always a convex cone. Here, $\backslash langle\; w,v\; \backslash rangle$ is the duality pairing between C and V, i.e. $\backslash langle\; w,v\backslash rangle\; =\; v(w)$.

More generally, the (algebraic) dual cone to C ⊂ V in a linear space V is a subset of the dual space V* defined by:

:$C^*\; :=\; \backslash left\; \backslash \{\; v\backslash in\; V^*\; \backslash mid\; \backslash forall\; w\backslash in\; C,\; v(w)\; \backslash ge\; 0\; \backslash right\; \backslash \}.$

In other words, if V* is the algebraic dual space of V, C* is the set of linear functionals that are nonnegative on the primal cone C. If we take V* to be the continuous dual space then it is the set of continuous linear functionals nonnegative on C.{{Cite book| url =https://books.google.com/books?id=oOYQVeHmNk4C|title=Applied Analysis|last1=Hunter|first1=John K.|last2=Nachtergaele|first2=Bruno|date=2001-01-01| publisher =World Scientific| isbn=9789810241919|pages=116|language=en}} This notion does not require the specification of an inner product on V.

In finite dimensions, the two notions of dual cone are essentially the same because every finite dimensional linear functional is continuous,{{Cite book|url=https://books.google.com/books?id=pec4r3EsIiMC|title=A Short Course on Banach Space Theory|last=Carothers|first=N. L.|date=2005-01-01 |publisher =Cambridge University Press|isbn=9780521603720|language=en}} and every continuous linear functional in an inner product space induces a linear isomorphism (nonsingular linear map) from V* to V, and this isomorphism will take the dual cone given by the second definition, in V*, onto the one given by the first definition; see the Riesz representation theorem.

If C is equal to its dual cone, then C is called self-dual. A cone can be said to be self-dual without reference to any given inner product, if there exists an inner product with respect to which it is equal to its dual by the first definition.

• Given a closed, convex subset K of Hilbert space V, the outward normal cone to the set K at the point x in K is given by
• : $N\_K(x)\; =\; \backslash left\backslash \{\; p\; \backslash in\; V\; \backslash colon\; \backslash forall\; x^*\; \backslash in\; K,\; \backslash left\backslash langle\; p,\; x^*\; -\; x\; \backslash right\backslash rangle\; \backslash leq\; 0\; \backslash right\backslash \}.$
• Given a closed, convex subset K of V, the tangent cone (or contingent cone) to the set K at the point x is given by
• : $T\_K(x)\; =\; \backslash overline\{\backslash bigcup\_\{h>0\}\; \backslash frac\{K\; -\; x\}\{h\}\}.$
• Given a closed, convex subset K of Hilbert space V, the tangent cone to the set K at the point x in K can be defined as polar cone to outwards normal cone $N\_K(x)$:
• : $T\_K(x)\; =\; N\_K^o(x)\; \backslash \; \backslash overset\{\backslash underset\{\backslash mathrm\{def\}\}\{\}\}\{=\}\backslash \; \backslash \{y\; \backslash in\; V\; \backslash mid\; \backslash forall\; \backslash xi\; \backslash in\; N\_K(x):\; \backslash langle\; y,\; \backslash xi\; \backslash rangle\; \backslash leqslant\; 0\; \backslash \}$

Both the normal and tangent cone have the property of being closed and convex.

They are important concepts in the fields of convex optimization, variational inequalities and projected dynamical systems.

If C is a non-empty convex cone in X, then the linear span of C is equal to C - C and the largest vector subspace of X contained in C is equal to C ∩ (−C).{{sfn |Narici|Beckenstein | 2011 | pp=149-153}}

A pointed and salient convex cone C induces a partial ordering "≥" on V, defined so that $x\backslash geq\; y$ if and only if $x-y\backslash in\; C.$ (If the cone is flat, the same definition gives merely a preorder.) Sums and positive scalar multiples of valid inequalities with respect to this order remain valid inequalities. A vector space with such an order is called an ordered vector space. Examples include the product order on real-valued vectors, $\backslash R^n,$ and the Loewner order on positive semidefinite matrices. Such an ordering is commonly found in semidefinite programming.

• Cone (disambiguation)
• Cone (geometry)
• Cone (topology)
• Farkas' lemma
• Bipolar theorem
• Ordered vector space

{{reflist|30em}}

• {{Cite book| last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Topological Vector Spaces | publisher=Springer-Verlag | location= Berlin, New York | series=Elements of Mathematics | isbn=978-3-540-13627-9 | year=1987 | url=https://books.google.com/books?id=S4wnAQAAIAAJ }}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{cite book | last=Rockafellar | first=R. T. | author-link=R. Tyrrell Rockafellar | title=Convex Analysis |publisher=Princeton University Press | location=Princeton, NJ | orig-year=1970 | year=1997 | isbn=1-4008-7317-7 |url=https://books.google.com/books?id=1TiOka9bx3sC }}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{cite book | last=Zălinescu | first=C. | title=Convex Analysis in General Vector Spaces | publisher=World Scientific | location= River Edge, NJ | year=2002 |isbn=981-238-067-1 | mr=1921556 | url=https://books.google.com/books?id=jcbUCgAAQBAJ }}

{{Functional analysis}}

{{Topological vector spaces}}

{{DEFAULTSORT:Convex Cone}}

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Category:Linear algebra