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

• If all ${\displaystyle x_{1},x_{2},\ldots }$  belong to ${\displaystyle S}$  then the convex series ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$  is called a convex series with elements of ${\displaystyle S}$ .
• If the set ${\displaystyle \left\{x_{1},x_{2},\ldots \right\}}$  is a (von Neumann) bounded set then the series called a b-convex series.
• The convex series ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$  is said to be a convergent series if the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}r_{i}x_{i}\right)_{n=1}^{\infty }}$  converges in ${\displaystyle X}$  to some element of ${\displaystyle X,}$  which is called the sum of the convex series.
• The convex series is called Cauchy if ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$  is a Cauchy series, which by definition means that the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}r_{i}x_{i}\right)_{n=1}^{\infty }}$  is a Cauchy 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 a topological 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, ${\displaystyle X}$  is not 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 a lcs-closed set if there exists a Fréchet 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 and convex (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 a li-convex set if there exists a Fréchet 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 ${\displaystyle S.}$ 

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

Conditions (Hx) and (Hwx) edit

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 ${\displaystyle \sum _{i=1}^{\infty }r_{i}(x_{i},y_{i})}$  is a convex series with elements of ${\displaystyle A}$  such that ${\displaystyle \sum _{i=1}^{\infty }r_{i}y_{i}}$  is convergent in ${\displaystyle Y}$  with sum ${\displaystyle y}$  and ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$  is Cauchy, then ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$  is convergent in ${\displaystyle X}$  and its sum ${\displaystyle x}$  is such that ${\displaystyle (x,y)\in A.}$ 
• Condition (Hwx): Whenever ${\displaystyle \sum _{i=1}^{\infty }r_{i}(x_{i},y_{i})}$  is a b-convex series with elements of ${\displaystyle A}$  such that ${\displaystyle \sum _{i=1}^{\infty }r_{i}y_{i}}$  is convergent in ${\displaystyle Y}$  with sum ${\displaystyle y}$  and ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$  is Cauchy, then ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$  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 ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$  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 ${\displaystyle {\mathcal {S}}:Y\rightrightarrows Z}$  are multifunctions and ${\displaystyle S\subseteq X}$  is a non-empty subset of a topological vector space ${\displaystyle X:}$ 

• The graph of a multifunction of ${\displaystyle {\mathcal {R}}}$  is the set ${\displaystyle \operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.}$ 
• ${\displaystyle {\mathcal {R}}}$  is closed (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 mulifunction ${\displaystyle {\mathcal {R}}}$  is convex if and only if for all ${\displaystyle x_{0},x_{1}\in X}$  and all ${\displaystyle r\in [0,1],}$  ${\displaystyle r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).}$ 
• The inverse 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 ${\displaystyle B\subseteq Y,}$  ${\displaystyle {\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).}$ 
• The domain of a multifunction ${\displaystyle {\mathcal {R}}}$  is ${\displaystyle \operatorname {Dom} {\mathcal {R}}:=\left\{x\in X:{\mathcal {R}}(x)\neq \emptyset \right\}.}$ 
• The image of a multifunction ${\displaystyle {\mathcal {R}}}$  is ${\displaystyle \operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x).}$  For any subset ${\displaystyle A\subseteq X,}$  ${\displaystyle {\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).}$ 
• The composition ${\displaystyle {\mathcal {S}}\circ {\mathcal {R}}:X\rightrightarrows Z}$  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 ${\displaystyle \implies }$  cs-complete ${\displaystyle \implies }$  cs-closed ${\displaystyle \implies }$  lower cs-closed (lcs-closed) and ideally convex.

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

(Hx) ${\displaystyle \implies }$  (Hwx) ${\displaystyle \implies }$  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 ${\displaystyle \operatorname {Pr} _{X}(A)}$  is bounded then,

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

Preserved properties edit

Let ${\displaystyle X_{0}}$  be a linear subspace of ${\displaystyle X.}$  Let ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$  and ${\displaystyle {\mathcal {S}}:Y\rightrightarrows Z}$  be multifunctions.

• 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.
• The Cartesian product of cs-closed (resp. ideally convex) subsets of arbitrarily many topological vector spaces has that same property (in the product space endowed with the product topology).
• The intersection of countably many lower ideally convex (resp. lower cs-closed) subsets of ${\displaystyle X}$  has the same property.
• The Cartesian 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 the product topology).
• Suppose ${\displaystyle X}$  is a Fréchet 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 a Fréchet 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 a Fréchet 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 ${\displaystyle X}$  is Hausdorff and finite dimensional then ${\displaystyle S}$  is cs-closed.
3. If ${\displaystyle X}$  is first countable and ${\displaystyle S}$  is ideally convex then ${\displaystyle \operatorname {int} S=\operatorname {int} \left(\operatorname {cl} S\right).}$ 

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

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

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

• 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 – via Internet 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.