In mathematics, particularly infunctional analysis andconvex analysis, aconvex series is aseries of the form${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$ where${\displaystyle x_1,x_2,\dots }$ are all elements of atopological vector space${\displaystyle X}$, and all${\displaystyle r_1,r_2,\dots }$ are non-negativereal numbers that sum to${\displaystyle 1}$ (that is, such that${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i=1}$).

Suppose that${\displaystyle S}$  is a subset of${\displaystyle X}$  and${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is a convex series in${\displaystyle X.}$ 

• If all${\displaystyle x_1,x_2,\dots }$  belong to${\displaystyle S}$  then the convex series${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is called aconvex series with elements of${\displaystyle S}$ .
• If the set${\displaystyle \left\{x_1,x_2,\dots \right\}}$  is a(von Neumann) bounded set then the series called ab-convex series.
• The convex series${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is said to be aconvergent series if the sequence of partial sums${\displaystyle {\left(\sum _{i=1}^n{r}_i{x}_i\right)}_{n=1}^{\mathrm{\infty }}}$  converges in${\displaystyle X}$  to some element of${\displaystyle X,}$  which is called thesum of the convex series.
• The convex series is calledCauchy if${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is aCauchy series, which by definition means that the sequence of partial sums${\displaystyle {\left(\sum _{i=1}^n{r}_i{x}_i\right)}_{n=1}^{\mathrm{\infty }}}$  is aCauchy sequence.

Convex series allow for the definition of special types of subsets that are well-behaved and useful with very good stability properties.

If${\displaystyle S}$  is a subset of atopological vector space${\displaystyle X}$  then${\displaystyle S}$  is said to be a:

• cs-closed set if any convergent convex series with elements of${\displaystyle S}$  has its (each) sum in${\displaystyle S.}$ 
  • In this definition,${\displaystyle X}$  isnot required to be Hausdorff, in which case the sum may not be unique. In any such case we require that every sum belong to${\displaystyle S.}$ 
• lower cs-closed set or alcs-closed set if there exists aFréchet space${\displaystyle Y}$  such that${\displaystyle S}$  is equal to the projection onto${\displaystyle X}$  (via the canonical projection) of some cs-closed subset${\displaystyle B}$  of${\displaystyle X\times Y}$  Every cs-closed set is lower cs-closed and every lower cs-closed set is lower ideally convex andconvex (the converses are not true in general).
• ideally convex set if any convergent b-series with elements of${\displaystyle S}$  has its sum in${\displaystyle S.}$ 
• lower ideally convex set or ali-convex set if there exists aFréchet space${\displaystyle Y}$  such that${\displaystyle S}$  is equal to the projection onto${\displaystyle X}$  (via the canonical projection) of some ideally convex subset${\displaystyle B}$  of${\displaystyle X\times Y.}$  Every ideally convex set is lower ideally convex. Every lower ideally convex set is convex but the converse is in general not true.
• cs-complete set if any Cauchy convex series with elements of${\displaystyle S}$  is convergent and its sum is in${\displaystyle S.}$ 
• bcs-complete set if any Cauchy b-convex series with elements of${\displaystyle S}$  is convergent and its sum is in${\displaystyle S.}$ 

Theempty set is convex, ideally convex, bcs-complete, cs-complete, and cs-closed.

If${\displaystyle X}$  and${\displaystyle Y}$  are topological vector spaces,${\displaystyle A}$  is a subset of${\displaystyle X\times Y,}$  and${\displaystyle x\in X}$  then${\displaystyle A}$  is said to satisfy:^[1]

• Condition (Hx): Whenever${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i(x_i,y_i)}$  is aconvex series with elements of${\displaystyle A}$  such that${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{y}_i}$  is convergent in${\displaystyle Y}$  with sum${\displaystyle y}$  and${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is Cauchy, then${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is convergent in${\displaystyle X}$  and its sum${\displaystyle x}$  is such that${\displaystyle (x,y)\in A.}$ 
• Condition (Hwx): Whenever${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i(x_i,y_i)}$  is ab-convex series with elements of${\displaystyle A}$  such that${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{y}_i}$  is convergent in${\displaystyle Y}$  with sum${\displaystyle y}$  and${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is Cauchy, then${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is convergent in${\displaystyle X}$  and its sum${\displaystyle x}$  is such that${\displaystyle (x,y)\in A.}$ 
  • If X is locally convex then the statement "and${\displaystyle \sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i}$  is Cauchy" may be removed from the definition of condition (Hwx).

The following notation and notions are used, where${\displaystyle \mathcal{R}:X\rightrightarrows Y}$  and${\displaystyle \mathcal{S}:Y\rightrightarrows Z}$  aremultifunctions and${\displaystyle S\subseteq X}$  is a non-empty subset of atopological vector space${\displaystyle X:}$ 

• Thegraph of a multifunction of${\displaystyle \mathcal{R}}$  is the set${\displaystyle \mathrm{gr}\mathcal{R}:=\{(x,y)\in X\times Y:y\in \mathcal{R}(x)\}.}$ 
• ${\displaystyle \mathcal{R}}$  isclosed (respectively,cs-closed,lower cs-closed,convex,ideally convex,lower ideally convex,cs-complete,bcs-complete) if the same is true of the graph of${\displaystyle \mathcal{R}}$  in${\displaystyle X\times Y.}$ 
  • The multifunction${\displaystyle \mathcal{R}}$  is convex if and only if for all${\displaystyle x_0,x_1\in X}$  and all${\displaystyle r\in [0,1],}$ ${\displaystyle r\mathcal{R}\left(x_0\right)+(1-r)\mathcal{R}\left(x_1\right)\subseteq \mathcal{R}\left(r{x}_0+(1-r)x_1\right).}$ 
• Theinverse of a multifunction${\displaystyle \mathcal{R}}$  is the multifunction${\displaystyle {\mathcal{R}}^{-1}:Y\rightrightarrows X}$  defined by${\displaystyle {\mathcal{R}}^{-1}(y):=\left\{x\in X:y\in \mathcal{R}(x)\right\}.}$  For any subset${\displaystyle B\subseteq Y,}$ ${\displaystyle {\mathcal{R}}^{-1}(B):={\cup }_{y\in B}{\mathcal{R}}^{-1}(y).}$ 
• Thedomain of a multifunction${\displaystyle \mathcal{R}}$  is${\displaystyle \mathrm{Dom}\mathcal{R}:=\left\{x\in X:\mathcal{R}(x)\ne \mathrm{\varnothing }\right\}.}$ 
• Theimage of a multifunction${\displaystyle \mathcal{R}}$  is${\displaystyle \mathrm{Im}\mathcal{R}:={\cup }_{x\in X}\mathcal{R}(x).}$  For any subset${\displaystyle A\subseteq X,}$ ${\displaystyle \mathcal{R}(A):={\cup }_{x\in A}\mathcal{R}(x).}$ 
• Thecomposition${\displaystyle \mathcal{S}\circ \mathcal{R}:X\rightrightarrows Z}$  is defined by${\displaystyle \left(\mathcal{S}\circ \mathcal{R}\right)(x):={\cup }_{y\in \mathcal{R}(x)}\mathcal{S}(y)}$  for each${\displaystyle x\in X.}$ 

Let${\displaystyle X,Y,\text{ and }Z}$  be topological vector spaces,${\displaystyle S\subseteq X,T\subseteq Y,}$  and${\displaystyle A\subseteq X\times Y.}$  The following implications hold:

complete${\displaystyle ⟹}$  cs-complete${\displaystyle ⟹}$  cs-closed${\displaystyle ⟹}$  lower cs-closed (lcs-closed)and ideally convex.

lower cs-closed (lcs-closed)or ideally convex${\displaystyle ⟹}$  lower ideally convex (li-convex)${\displaystyle ⟹}$  convex.

(Hx)${\displaystyle ⟹}$  (Hwx)${\displaystyle ⟹}$  convex.

The converse implications do not hold in general.

If${\displaystyle X}$  is complete then,

1. ${\displaystyle S}$  is cs-complete (respectively, bcs-complete) if and only if${\displaystyle S}$  is cs-closed (respectively, ideally convex).
2. ${\displaystyle A}$  satisfies (Hx) if and only if${\displaystyle A}$  is cs-closed.
3. ${\displaystyle A}$  satisfies (Hwx) if and only if${\displaystyle A}$  is ideally convex.

If${\displaystyle Y}$  is complete then,

1. ${\displaystyle A}$  satisfies (Hx) if and only if${\displaystyle A}$  is cs-complete.
2. ${\displaystyle A}$  satisfies (Hwx) if and only if${\displaystyle A}$  is bcs-complete.
3. If${\displaystyle B\subseteq X\times Y\times Z}$  and${\displaystyle y\in Y}$  then:
   1. ${\displaystyle B}$  satisfies (H(x, y)) if and only if${\displaystyle B}$  satisfies (Hx).
   2. ${\displaystyle B}$  satisfies (Hw(x, y)) if and only if${\displaystyle B}$  satisfies (Hwx).

If${\displaystyle X}$  is locally convex and${\displaystyle {\mathrm{Pr}}_X(A)}$  is bounded then,

1. If${\displaystyle A}$  satisfies (Hx) then${\displaystyle {\mathrm{Pr}}_X(A)}$  is cs-closed.
2. If${\displaystyle A}$  satisfies (Hwx) then${\displaystyle {\mathrm{Pr}}_X(A)}$  is ideally convex.

Let${\displaystyle X_0}$  be a linear subspace of${\displaystyle X.}$  Let${\displaystyle \mathcal{R}:X\rightrightarrows Y}$  and${\displaystyle \mathcal{S}:Y\rightrightarrows Z}$  bemultifunctions.

• If${\displaystyle S}$  is a cs-closed (resp. ideally convex) subset of${\displaystyle X}$  then${\displaystyle X_0\cap S}$  is also a cs-closed (resp. ideally convex) subset of${\displaystyle X_0.}$ 
• If${\displaystyle X}$  is first countable then${\displaystyle X_0}$  is cs-closed (resp. cs-complete) if and only if${\displaystyle X_0}$  is closed (resp. complete); moreover, if${\displaystyle X}$  is locally convex then${\displaystyle X_0}$  is closed if and only if${\displaystyle X_0}$  is ideally convex.
• ${\displaystyle S\times T}$  is cs-closed (resp. cs-complete, ideally convex, bcs-complete) in${\displaystyle X\times Y}$  if and only if the same is true of both${\displaystyle S}$  in${\displaystyle X}$  and of${\displaystyle T}$  in${\displaystyle Y.}$ 
• The properties of being cs-closed, lower cs-closed, ideally convex, lower ideally convex, cs-complete, and bcs-complete are all preserved under isomorphisms of topological vector spaces.
• The intersection of arbitrarily many cs-closed (resp. ideally convex) subsets of${\displaystyle X}$  has the same property.
• TheCartesian product of cs-closed (resp. ideally convex) subsets of arbitrarily many topological vector spaces has that same property (in the product space endowed with theproduct topology).
• The intersection of countably many lower ideally convex (resp. lower cs-closed) subsets of${\displaystyle X}$  has the same property.
• TheCartesian product of lower ideally convex (resp. lower cs-closed) subsets of countably many topological vector spaces has that same property (in the product space endowed with theproduct topology).
• Suppose${\displaystyle X}$  is aFréchet space and the${\displaystyle A}$  and${\displaystyle B}$  are subsets. If${\displaystyle A}$  and${\displaystyle B}$  are lower ideally convex (resp. lower cs-closed) then so is${\displaystyle A+B.}$ 
• Suppose${\displaystyle X}$  is aFréchet space and${\displaystyle A}$  is a subset of${\displaystyle X.}$  If${\displaystyle A}$  and${\displaystyle \mathcal{R}:X\rightrightarrows Y}$  are lower ideally convex (resp. lower cs-closed) then so is${\displaystyle \mathcal{R}(A).}$ 
• Suppose${\displaystyle Y}$  is aFréchet space and${\displaystyle {\mathcal{R}}_2:X\rightrightarrows Y}$  is a multifunction. If${\displaystyle \mathcal{R},{\mathcal{R}}_2,\mathcal{S}}$  are all lower ideally convex (resp. lower cs-closed) then so are${\displaystyle \mathcal{R}+{\mathcal{R}}_2:X\rightrightarrows Y}$  and${\displaystyle \mathcal{S}\circ \mathcal{R}:X\rightrightarrows Z.}$ 

If${\displaystyle S}$  be a non-empty convex subset of a topological vector space${\displaystyle X}$  then,

1. If${\displaystyle S}$  is closed or open then${\displaystyle S}$  is cs-closed.
2. If${\displaystyle X}$  isHausdorff and finite dimensional then${\displaystyle S}$  is cs-closed.
3. If${\displaystyle X}$  isfirst countable and${\displaystyle S}$  is ideally convex then${\displaystyle \mathrm{int}S=\mathrm{int}\left(\mathrm{cl}S\right).}$ 

Let${\displaystyle X}$  be aFréchet space,${\displaystyle Y}$  be a topological vector spaces,${\displaystyle A\subseteq X\times Y,}$  and${\displaystyle {\mathrm{Pr}}_Y:X\times Y\to Y}$  be the canonical projection. If${\displaystyle A}$  is lower ideally convex (resp. lower cs-closed) then the same is true of${\displaystyle {\mathrm{Pr}}_Y(A).}$ 

If${\displaystyle X}$  is a barreledfirst countable space and if${\displaystyle C\subseteq X}$  then:

1. If${\displaystyle C}$  is lower ideally convex then${\displaystyle C^i=\mathrm{int}C,}$  where${\displaystyle C^i:={\mathrm{aint}}_{X}C}$  denotes thealgebraic interior of${\displaystyle C}$  in${\displaystyle X.}$ 
2. If${\displaystyle C}$  is ideally convex then${\displaystyle C^i=\mathrm{int}C=\mathrm{int}\left(\mathrm{cl}C\right)={\left(\mathrm{cl}C\right)}^i.}$ 

• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

• Zălinescu, Constantin (30 July 2002).Convex Analysis in General Vector Spaces. River Edge, N.J. London:World Scientific Publishing.ISBN 978-981-4488-15-0.MR 1921556.OCLC 285163112 – viaInternet Archive.
• Baggs, Ivan (1974)."Functions with a closed graph".Proceedings of the American Mathematical Society.43 (2):439–442.doi:10.1090/S0002-9939-1974-0334132-8.ISSN 0002-9939.