Mgkrupa: Added item
In mathematics, particularly in [[functional analysis]] and [[convex analysis]], a ''convex series'' is a [[Series (mathematics)|series]] of the form <math>\sum_{i=1}^{\infty} r_i x_i</math> where <math>x_1, x_2, \ldots</math> are all elements of a [[topological vector space]] ''X'', all <math>r_1, r_2, \ldots</math> are non-negative [[real numbers]] that sum to ''1'' (i.e. <math>\sum_{i=1}^{\infty} r_i = 1</math>).
==Types of Convex series==
Suppose that ''S'' is a subset of ''X'' and <math>\sum_{i=1}^{\infty} r_i x_i</math> is a convex series in ''X''.
* If all <math>x_1, x_2, \ldots</math> belong to ''S'' then we say that the convex series <math>\sum_{i=1}^{\infty} r_i x_i</math> is a '''convex series with elements of ''S'''''.
* If the set <math>\{ x_1, x_2, \ldots \} </math> is [[Bounded set (topological vector space)|von Neumann bounded]] then we call the series a '''b-convex series'''.
* We say that the convex series is '''convergent''' if <math>\sum_{i=1}^{\infty} r_i x_i</math> converges in ''X'' (i.e. if the sequence of partial sums <math>\left( \sum_{i=1}^{n} r_i x_i \right)_{n=1}^{\infty}</math> converges in ''X'' to some element of ''X'', which is called the convex series' '''sum''').
* We call the convex series '''Cauchy''' if <math>\sum_{i=1}^{\infty} r_i x_i</math> is a Cauchy series (i.e. if the sequence of partial sums <math>\left( \sum_{i=1}^{n} r_i x_i \right)_{n=1}^{\infty}</math> is a [[Cauchy sequence]]).
==Subsets==
Convex series allow for the definition of special types of subsets that are well-behaved and useful with very good stability properties.
Let ''S'' be a subset of a [[topological vector space]] ''X''. Then say that ''S'' is:
* '''cs-closed''' if any convergent convex series with elements of ''S'' has its (each) sum in ''S''.
** Note that ''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 ''S''.
* '''lower cs-closed''' or '''lcs-closed''' if there exists a [[Fréchet space]] ''Y'' such that ''S'' is equal to the projection onto ''X'' (via the canonical projection) of some cs-closed subset ''B'' of <math>X \times Y</math> Every cs-closed set is lower cs-closed and every lower cs-closed set is lower ideally convex and [[convex set|convex]] (the converses are not true in general).
* '''ideally convex''' if any convergent b-series with elements of ''S'' has its sum in ''S''.
* '''lower ideally convex''' or '''li-convex''' if there exists a [[Fréchet space]] ''Y'' such that ''S'' is equal to the projection onto ''X'' (via the canonical projection) of some ideally convex subset ''B'' of <math>X \times Y</math>. Every ideally convex set is lower ideally convex. Every lower ideally convex set is [[convex set|convex]] but the converse is in general not true.
* '''cs-complete''' if any Cauchy convex series with elements of ''S'' is convergent and its sum is in ''S''.
* '''bcs-complete''' if any Cauchy b-convex series with elements of ''S'' is convergent and its sum is in ''S''.
Note that the [[empty set]] is convex, ideally convex, bcs-complete, cs-complete, and cs-closed.
===Conditions (Hx) and (Hwx)===
If ''X'' and ''Y'' are topological vector spaces, ''A'' is a subset of <math>X \times Y</math>, and ''x'' is an element of ''X'' then we say that ''A'' satisfies:
* '''Condition (H''x'')''': Whenever <math>\sum_{i=1}^{\infty} r_i (x_i, y_i)</math> is a ''convex'' series with elements of ''A'' such that <math>\sum_{i=1}^{\infty} r_i y_i</math> is convergent in ''Y'' with sum ''y'' and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy, then <math>\sum_{i=1}^{\infty} r_i x_i</math> is convergent in ''X'' and its sum ''x'' is such that <math>(x, y) \in A</math>.
* '''Condition (Hw''x'')''': Whenever <math>\sum_{i=1}^{\infty} r_i (x_i, y_i)</math> is a ''b-convex'' series with elements of ''A'' such that <math>\sum_{i=1}^{\infty} r_i y_i</math> is convergent in ''Y'' with sum ''y'' and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy, then <math>\sum_{i=1}^{\infty} r_i x_i</math> is convergent in ''X'' and its sum ''x'' is such that <math>(x, y) \in A</math>.
** Note that if X is locally convex then the statement "and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy" may be removed from the definition of condition (Hw''x'').
==Multifunctions==
The following notation and notions are used, where <math>\mathcal{R} : X \rightrightarrows Y</math> and <math>\mathcal{S} : Y \rightrightarrows Z</math> be [[multifunction]]s and ''S'' is a non-empty subset of a [[topological vector space]] ''X'':
* The '''graph of <math>\mathcal{R}</math>''' is <math>\operatorname{gr} \mathcal{R} := \left\{ (x, y) \in X \times Y : y \in \mathcal{R}(x) \right\}</math>.
* The '''inverse of <math>\mathcal{R}</math>''' is the multifunction <math>\mathcal{R}^{-1} : Y \rightrightarrows X</math> defined by <math>\mathcal{R}^{-1}(y) := \left\{ x \in X : y \in \mathcal{R}(x) \right\}</math>. For any subset <math>B \subseteq Y</math>, <math>\mathcal{R}^{-1}(B) := \cup_{y \in B} \mathcal{R}^{-1}(y)</math>.
* <math>\mathcal{R}</math> 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 <math>\mathcal{R}</math> in <math>X \times Y</math>.
** Note that <math>\mathcal{R}</math> is convex if and only if for all <math>x_0, x_1 \in X</math> and all <math>r \in [0, 1]</math>, <math>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)</math>.
* The '''domain of <math>\mathcal{R}</math>''' is <math>\operatorname{Dom} \mathcal{R} := \left\{ x \in X : \mathcal{R}(x) \neq \emptyset \right\}</math>.
* The '''image of <math>\mathcal{R}</math>''' is <math>\operatorname{Im} \mathcal{R} := \cup_{x \in X} \mathcal{R}(x)</math>. For any subset <math>A \subseteq X</math>, <math>\mathcal{R}(A) := \cup_{x \in A} \mathcal{R}(x)</math>.
* The composition <math>\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z</math> is defined by <math>\left( \mathcal{S} \circ \mathcal{R} \right)(x) := \cup_{y \in \mathcal{R}(x)} \mathcal{S}(y)</math> for each <math>x \in X</math>.
==Relationships==
Let ''X'',''Y'', and ''Z'' be topological vector spaces, <math>S \subseteq X</math>, <math>T \subseteq Y</math>, and <math>A \subseteq X \times Y</math>. The following implications hold:
:complete <math>\implies</math> cs-complete <math>\implies</math> cs-closed <math>\implies</math> lower cs-closed (lcs-closed) ''and'' ideally convex.
:lower cs-closed (lcs-closed) ''or'' ideally convex <math>\implies</math> lower ideally convex (li-convex) <math>\implies</math> convex.
:(H''x'') <math>\implies</math> (Hw''x'') <math>\implies</math> convex.
The converse implication do not hold in general.
If ''X'' is complete then,
# ''S'' is cs-complete (resp. bcs-complete) if and only if ''S'' is cs-closed (resp. ideally convex).
# ''A'' satisfies (H''x'') if and only if ''A'' is cs-closed.
# ''A'' satisfies (Hw''x'') if and only if ''A'' is ideally convex.
If ''Y'' is complete then,
# ''A'' satisfies (H''x'') if and only if ''A'' is cs-complete.
# ''A'' satisfies (Hw''x'') if and only if ''A'' is bcs-complete.
# If <math>B \subseteq X \times Y \times Z</math> and <math>y \in Y</math> then:
## ''B'' satisfies (H''(x, y)'') if and only if ''B'' satisfies (H''x'').
## ''B'' satisfies (Hw''(x, y)'') if and only if ''B'' satisfies (Hw''x'').
If ''X'' is locally convex and <math>\operatorname{Pr}_X (A)</math> is bounded then,
# If ''A'' satisfies (H''x'') then <math>\operatorname{Pr}_X (A)</math> is cs-closed.
# If ''A'' satisfies (Hw''x'') then <math>\operatorname{Pr}_X (A)</math> is ideally convex.
===Preserved properties===
Let <math>X_0</math> be a linear subspace of ''X''. Let <math>\mathcal{R} : X \rightrightarrows Y</math> and <math>\mathcal{S} : Y \rightrightarrows Z</math> be [[multifunction]]s.
* If ''S'' is a cs-closed (resp. ideally convex) subset of ''X'' then <math>X_0 \cap S</math> is also a cs-closed (resp. ideally convex) subset of <math>X_0</math>.
* If ''X'' is first countable then <math>X_0</math> is cs-closed (resp. cs-complete) if and only if <math>X_0</math> is closed (resp. complete); moreover, if ''X'' is locally convex then <math>X_0</math> is closed if and only if <math>X_0</math> is ideally convex.
* <math>S \times T</math> is cs-closed (resp. cs-complete, ideally convex, bcs-complete) in <math>X \times Y</math> if and only if the same is true of both ''S'' in ''X'' and of ''T'' in ''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 ''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 ''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 ''X'' is a [[Fréchet space]] and the ''A'' and ''B'' are subsets. If ''A'' and ''B'' are lower ideally convex (resp. lower cs-closed) then so is ''A + B''.
* Suppose ''X'' is a [[Fréchet space]] and ''A'' is a subset of ''X''. If ''A'' and <math>\mathcal{R} : X \rightrightarrows Y</math> are lower ideally convex (resp. lower cs-closed) then so is <math>\mathcal{R}(A)</math>.
* Suppose ''Y'' is a [[Fréchet space]] and <math>\mathcal{R}_2 : X \rightrightarrows Y</math> is a multifunction. If <math>\mathcal{R}, \mathcal{R}_2, \mathcal{S}</math> are all lower ideally convex (resp. lower cs-closed) then so are <math>\mathcal{R} + \mathcal{R}_2 : X \rightrightarrows Y</math> and <math>\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z</math>.
==Properties==
Let ''S'' be a non-empty convex subset of a topological vector space ''X''. Then,
# If ''S'' is closed or open then ''S'' is cs-closed.
# If ''X'' is [[Hausdorff]] and finite dimensional then ''S'' is cs-closed.
# If ''X'' is [[first countable]] and ''S'' is ideally convex then <math>\operatorname{int} S = \operatorname{int} \left( \operatorname{cl} S \right)</math>.
Let ''X'' be a [[Fréchet space]], ''Y'' be a topological vector spaces, <math>A \subseteq X \times Y</math>, and <math>\operatorname{Pr}_Y : X \times Y \to Y</math> be the canonical projection. If ''A'' is lower ideally convex (resp. lower cs-closed) then the same is true of <math>\operatorname{Pr}_Y (A)</math>.
Let ''X'' be a barreled [[first countable]] space and let ''C'' be a subset of ''X''. Then:
# If ''C'' is lower ideally convex then <math>C^{i} = \operatorname{int} C</math>, where <math>C^{i} := \operatorname{aint}_X C</math> denotes the [[algebraic interior]] of ''C'' in ''X''.
# If ''C'' is ideally convex then <math>C^{i} = \operatorname{int} C = \operatorname{int} \left( \operatorname{cl} C \right) = \left( \operatorname{cl} C\right)^i</math>.
==See also==
* [[convex]]
* [[Ursescu theorem]]
==Notes==
== References ==
*
* Liquid error: wrong number of arguments (given 1, expected 2)
[[Category:Theorems in functional analysis]]
==Types of Convex series==
Suppose that ''S'' is a subset of ''X'' and <math>\sum_{i=1}^{\infty} r_i x_i</math> is a convex series in ''X''.
* If all <math>x_1, x_2, \ldots</math> belong to ''S'' then we say that the convex series <math>\sum_{i=1}^{\infty} r_i x_i</math> is a '''convex series with elements of ''S'''''.
* If the set <math>\{ x_1, x_2, \ldots \} </math> is [[Bounded set (topological vector space)|von Neumann bounded]] then we call the series a '''b-convex series'''.
* We say that the convex series is '''convergent''' if <math>\sum_{i=1}^{\infty} r_i x_i</math> converges in ''X'' (i.e. if the sequence of partial sums <math>\left( \sum_{i=1}^{n} r_i x_i \right)_{n=1}^{\infty}</math> converges in ''X'' to some element of ''X'', which is called the convex series' '''sum''').
* We call the convex series '''Cauchy''' if <math>\sum_{i=1}^{\infty} r_i x_i</math> is a Cauchy series (i.e. if the sequence of partial sums <math>\left( \sum_{i=1}^{n} r_i x_i \right)_{n=1}^{\infty}</math> is a [[Cauchy sequence]]).
==Subsets==
Convex series allow for the definition of special types of subsets that are well-behaved and useful with very good stability properties.
Let ''S'' be a subset of a [[topological vector space]] ''X''. Then say that ''S'' is:
* '''cs-closed''' if any convergent convex series with elements of ''S'' has its (each) sum in ''S''.
** Note that ''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 ''S''.
* '''lower cs-closed''' or '''lcs-closed''' if there exists a [[Fréchet space]] ''Y'' such that ''S'' is equal to the projection onto ''X'' (via the canonical projection) of some cs-closed subset ''B'' of <math>X \times Y</math> Every cs-closed set is lower cs-closed and every lower cs-closed set is lower ideally convex and [[convex set|convex]] (the converses are not true in general).
* '''ideally convex''' if any convergent b-series with elements of ''S'' has its sum in ''S''.
* '''lower ideally convex''' or '''li-convex''' if there exists a [[Fréchet space]] ''Y'' such that ''S'' is equal to the projection onto ''X'' (via the canonical projection) of some ideally convex subset ''B'' of <math>X \times Y</math>. Every ideally convex set is lower ideally convex. Every lower ideally convex set is [[convex set|convex]] but the converse is in general not true.
* '''cs-complete''' if any Cauchy convex series with elements of ''S'' is convergent and its sum is in ''S''.
* '''bcs-complete''' if any Cauchy b-convex series with elements of ''S'' is convergent and its sum is in ''S''.
Note that the [[empty set]] is convex, ideally convex, bcs-complete, cs-complete, and cs-closed.
===Conditions (Hx) and (Hwx)===
If ''X'' and ''Y'' are topological vector spaces, ''A'' is a subset of <math>X \times Y</math>, and ''x'' is an element of ''X'' then we say that ''A'' satisfies:
* '''Condition (H''x'')''': Whenever <math>\sum_{i=1}^{\infty} r_i (x_i, y_i)</math> is a ''convex'' series with elements of ''A'' such that <math>\sum_{i=1}^{\infty} r_i y_i</math> is convergent in ''Y'' with sum ''y'' and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy, then <math>\sum_{i=1}^{\infty} r_i x_i</math> is convergent in ''X'' and its sum ''x'' is such that <math>(x, y) \in A</math>.
* '''Condition (Hw''x'')''': Whenever <math>\sum_{i=1}^{\infty} r_i (x_i, y_i)</math> is a ''b-convex'' series with elements of ''A'' such that <math>\sum_{i=1}^{\infty} r_i y_i</math> is convergent in ''Y'' with sum ''y'' and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy, then <math>\sum_{i=1}^{\infty} r_i x_i</math> is convergent in ''X'' and its sum ''x'' is such that <math>(x, y) \in A</math>.
** Note that if X is locally convex then the statement "and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy" may be removed from the definition of condition (Hw''x'').
==Multifunctions==
The following notation and notions are used, where <math>\mathcal{R} : X \rightrightarrows Y</math> and <math>\mathcal{S} : Y \rightrightarrows Z</math> be [[multifunction]]s and ''S'' is a non-empty subset of a [[topological vector space]] ''X'':
* The '''graph of <math>\mathcal{R}</math>''' is <math>\operatorname{gr} \mathcal{R} := \left\{ (x, y) \in X \times Y : y \in \mathcal{R}(x) \right\}</math>.
* The '''inverse of <math>\mathcal{R}</math>''' is the multifunction <math>\mathcal{R}^{-1} : Y \rightrightarrows X</math> defined by <math>\mathcal{R}^{-1}(y) := \left\{ x \in X : y \in \mathcal{R}(x) \right\}</math>. For any subset <math>B \subseteq Y</math>, <math>\mathcal{R}^{-1}(B) := \cup_{y \in B} \mathcal{R}^{-1}(y)</math>.
* <math>\mathcal{R}</math> 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 <math>\mathcal{R}</math> in <math>X \times Y</math>.
** Note that <math>\mathcal{R}</math> is convex if and only if for all <math>x_0, x_1 \in X</math> and all <math>r \in [0, 1]</math>, <math>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)</math>.
* The '''domain of <math>\mathcal{R}</math>''' is <math>\operatorname{Dom} \mathcal{R} := \left\{ x \in X : \mathcal{R}(x) \neq \emptyset \right\}</math>.
* The '''image of <math>\mathcal{R}</math>''' is <math>\operatorname{Im} \mathcal{R} := \cup_{x \in X} \mathcal{R}(x)</math>. For any subset <math>A \subseteq X</math>, <math>\mathcal{R}(A) := \cup_{x \in A} \mathcal{R}(x)</math>.
* The composition <math>\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z</math> is defined by <math>\left( \mathcal{S} \circ \mathcal{R} \right)(x) := \cup_{y \in \mathcal{R}(x)} \mathcal{S}(y)</math> for each <math>x \in X</math>.
==Relationships==
Let ''X'',''Y'', and ''Z'' be topological vector spaces, <math>S \subseteq X</math>, <math>T \subseteq Y</math>, and <math>A \subseteq X \times Y</math>. The following implications hold:
:complete <math>\implies</math> cs-complete <math>\implies</math> cs-closed <math>\implies</math> lower cs-closed (lcs-closed) ''and'' ideally convex.
:lower cs-closed (lcs-closed) ''or'' ideally convex <math>\implies</math> lower ideally convex (li-convex) <math>\implies</math> convex.
:(H''x'') <math>\implies</math> (Hw''x'') <math>\implies</math> convex.
The converse implication do not hold in general.
If ''X'' is complete then,
# ''S'' is cs-complete (resp. bcs-complete) if and only if ''S'' is cs-closed (resp. ideally convex).
# ''A'' satisfies (H''x'') if and only if ''A'' is cs-closed.
# ''A'' satisfies (Hw''x'') if and only if ''A'' is ideally convex.
If ''Y'' is complete then,
# ''A'' satisfies (H''x'') if and only if ''A'' is cs-complete.
# ''A'' satisfies (Hw''x'') if and only if ''A'' is bcs-complete.
# If <math>B \subseteq X \times Y \times Z</math> and <math>y \in Y</math> then:
## ''B'' satisfies (H''(x, y)'') if and only if ''B'' satisfies (H''x'').
## ''B'' satisfies (Hw''(x, y)'') if and only if ''B'' satisfies (Hw''x'').
If ''X'' is locally convex and <math>\operatorname{Pr}_X (A)</math> is bounded then,
# If ''A'' satisfies (H''x'') then <math>\operatorname{Pr}_X (A)</math> is cs-closed.
# If ''A'' satisfies (Hw''x'') then <math>\operatorname{Pr}_X (A)</math> is ideally convex.
===Preserved properties===
Let <math>X_0</math> be a linear subspace of ''X''. Let <math>\mathcal{R} : X \rightrightarrows Y</math> and <math>\mathcal{S} : Y \rightrightarrows Z</math> be [[multifunction]]s.
* If ''S'' is a cs-closed (resp. ideally convex) subset of ''X'' then <math>X_0 \cap S</math> is also a cs-closed (resp. ideally convex) subset of <math>X_0</math>.
* If ''X'' is first countable then <math>X_0</math> is cs-closed (resp. cs-complete) if and only if <math>X_0</math> is closed (resp. complete); moreover, if ''X'' is locally convex then <math>X_0</math> is closed if and only if <math>X_0</math> is ideally convex.
* <math>S \times T</math> is cs-closed (resp. cs-complete, ideally convex, bcs-complete) in <math>X \times Y</math> if and only if the same is true of both ''S'' in ''X'' and of ''T'' in ''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 ''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 ''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 ''X'' is a [[Fréchet space]] and the ''A'' and ''B'' are subsets. If ''A'' and ''B'' are lower ideally convex (resp. lower cs-closed) then so is ''A + B''.
* Suppose ''X'' is a [[Fréchet space]] and ''A'' is a subset of ''X''. If ''A'' and <math>\mathcal{R} : X \rightrightarrows Y</math> are lower ideally convex (resp. lower cs-closed) then so is <math>\mathcal{R}(A)</math>.
* Suppose ''Y'' is a [[Fréchet space]] and <math>\mathcal{R}_2 : X \rightrightarrows Y</math> is a multifunction. If <math>\mathcal{R}, \mathcal{R}_2, \mathcal{S}</math> are all lower ideally convex (resp. lower cs-closed) then so are <math>\mathcal{R} + \mathcal{R}_2 : X \rightrightarrows Y</math> and <math>\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z</math>.
==Properties==
Let ''S'' be a non-empty convex subset of a topological vector space ''X''. Then,
# If ''S'' is closed or open then ''S'' is cs-closed.
# If ''X'' is [[Hausdorff]] and finite dimensional then ''S'' is cs-closed.
# If ''X'' is [[first countable]] and ''S'' is ideally convex then <math>\operatorname{int} S = \operatorname{int} \left( \operatorname{cl} S \right)</math>.
Let ''X'' be a [[Fréchet space]], ''Y'' be a topological vector spaces, <math>A \subseteq X \times Y</math>, and <math>\operatorname{Pr}_Y : X \times Y \to Y</math> be the canonical projection. If ''A'' is lower ideally convex (resp. lower cs-closed) then the same is true of <math>\operatorname{Pr}_Y (A)</math>.
Let ''X'' be a barreled [[first countable]] space and let ''C'' be a subset of ''X''. Then:
# If ''C'' is lower ideally convex then <math>C^{i} = \operatorname{int} C</math>, where <math>C^{i} := \operatorname{aint}_X C</math> denotes the [[algebraic interior]] of ''C'' in ''X''.
# If ''C'' is ideally convex then <math>C^{i} = \operatorname{int} C = \operatorname{int} \left( \operatorname{cl} C \right) = \left( \operatorname{cl} C\right)^i</math>.
==See also==
* [[convex]]
* [[Ursescu theorem]]
==Notes==
== References ==
*
* Liquid error: wrong number of arguments (given 1, expected 2)
[[Category:Theorems in functional analysis]]
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