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Ursescu theorem
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<h2 id="ursescu-theorem">Ursescu theorem</h2> <p>The following notation and notions are used, where R : X ⇉ Y {\displaystyle {\mathcal {R}}:X\rightrightarrows Y} is a <a href="/facts/Set-valued_function/WH5ua6BL">set-valued function</a> and S {\displaystyle S} is a non-empty subset of a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> X {\displaystyle X} : </p> <ul><li>the <a href="/facts/Affine_span/4fhyHKoa">affine span</a> of S {\displaystyle S} is denoted by aff S {\displaystyle \operatorname {aff} S} and the <a href="/facts/Linear_span/vPjclEtb">linear span</a> is denoted by span S . {\displaystyle \operatorname {span} S.} </li> <li> S i := aint X S {\displaystyle S^{i}:=\operatorname {aint} _{X}S} denotes the <a href="/facts/Algebraic_interior/oCWBqUr9">algebraic interior</a> of S {\displaystyle S} in X . {\displaystyle X.} </li> <li> i S := aint aff ( S − S ) S {\displaystyle {}^{i}S:=\operatorname {aint} _{\operatorname {aff} (S-S)}S} denotes the <a href="/facts/Algebraic_interior/oCWBqUr9">relative algebraic interior</a> of S {\displaystyle S} (i.e. the algebraic interior of S {\displaystyle S} in aff ( S − S ) {\displaystyle \operatorname {aff} (S-S)} ).</li> <li> i b S := i S {\displaystyle {}^{ib}S:={}^{i}S} if span ( S − s 0 ) {\displaystyle \operatorname {span} \left(S-s_{0}\right)} is <a href="/facts/Barrelled_space/r2H58TN3">barreled</a> for some/every s 0 ∈ S {\displaystyle s_{0}\in S} while i b S := ∅ {\displaystyle {}^{ib}S:=\varnothing } otherwise. <ul><li>If S {\displaystyle S} is convex then it can be shown that for any x ∈ X , {\displaystyle x\in X,} x ∈ i b S {\displaystyle x\in {}^{ib}S} if and only if the cone generated by S − x {\displaystyle S-x} is a barreled linear subspace of X {\displaystyle X} or equivalently, if and only if ∪ n ∈ N n ( S − x ) {\displaystyle \cup _{n\in \mathbb {N} }n(S-x)} is a barreled linear subspace of X {\displaystyle X} </li></ul></li> <li>The domain of R {\displaystyle {\mathcal {R}}} is Dom R := { x ∈ X : R ( x ) ≠ ∅ } . {\displaystyle \operatorname {Dom} {\mathcal {R}}:=\{x\in X:{\mathcal {R}}(x)\neq \varnothing \}.} </li> <li>The image of R {\displaystyle {\mathcal {R}}} is Im R := ∪ x ∈ X R ( x ) . {\displaystyle \operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x).} For any subset A ⊆ X , {\displaystyle A\subseteq X,} R ( A ) := ∪ x ∈ A R ( x ) . {\displaystyle {\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).} </li> <li>The graph of R {\displaystyle {\mathcal {R}}} is gr R := { ( x , y ) ∈ X × Y : y ∈ R ( x ) } . {\displaystyle \operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.} </li> <li> R {\displaystyle {\mathcal {R}}} is closed (respectively, convex) if the graph of R {\displaystyle {\mathcal {R}}} is closed (resp. convex) in X × Y . {\displaystyle X\times Y.} <ul><li>Note that R {\displaystyle {\mathcal {R}}} is convex if and only if for all x 0 , x 1 ∈ X {\displaystyle x_{0},x_{1}\in X} and all r ∈ [ 0 , 1 ] , {\displaystyle r\in [0,1],} r R ( x 0 ) + ( 1 − r ) R ( x 1 ) ⊆ R ( r x 0 + ( 1 − r ) x 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).} </li></ul></li> <li>The inverse of R {\displaystyle {\mathcal {R}}} is the set-valued function R − 1 : Y ⇉ X {\displaystyle {\mathcal {R}}^{-1}:Y\rightrightarrows X} defined by R − 1 ( y ) := { x ∈ X : y ∈ R ( x ) } . {\displaystyle {\mathcal {R}}^{-1}(y):=\{x\in X:y\in {\mathcal {R}}(x)\}.} For any subset B ⊆ Y , {\displaystyle B\subseteq Y,} R − 1 ( B ) := ∪ y ∈ B R − 1 ( y ) . {\displaystyle {\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).} <ul><li>If f : X → Y {\displaystyle f:X\to Y} is a function, then its inverse is the set-valued function f − 1 : Y ⇉ X {\displaystyle f^{-1}:Y\rightrightarrows X} obtained from canonically identifying f {\displaystyle f} with the set-valued function f : X ⇉ Y {\displaystyle f:X\rightrightarrows Y} defined by x ↦ { f ( x ) } . {\displaystyle x\mapsto \{f(x)\}.} </li></ul></li> <li> int T S {\displaystyle \operatorname {int} _{T}S} is the <a href="/facts/Interior_(topology)/5sxGnQfY">topological interior</a> of S {\displaystyle S} with respect to T , {\displaystyle T,} where S ⊆ T . {\displaystyle S\subseteq T.} </li> <li> rint S := int aff S S {\displaystyle \operatorname {rint} S:=\operatorname {int} _{\operatorname {aff} S}S} is the <a href="/facts/Interior_(topology)/5sxGnQfY">interior</a> of S {\displaystyle S} with respect to aff S . {\displaystyle \operatorname {aff} S.} </li></ul> <h3>Statement</h3> <p>Theorem<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> (Ursescu)—Let X {\displaystyle X} be a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a> <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">semi-metrizable</a> <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> and R : X ⇉ Y {\displaystyle {\mathcal {R}}:X\rightrightarrows Y} be a <a href="/facts/Closed_(mathematics)/upYnRTUj">closed</a> <a href="/facts/Convex_function/IbzG5SLF">convex</a> multifunction with non-empty domain. Assume that span ( Im R − y ) {\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)} is a <a href="/facts/Barrelled_space/r2H58TN3">barrelled space</a> for some/every y ∈ Im R . {\displaystyle y\in \operatorname {Im} {\mathcal {R}}.} Assume that y 0 ∈ i ( Im R ) {\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})} and let x 0 ∈ R − 1 ( y 0 ) {\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)} (so that y 0 ∈ R ( x 0 ) {\displaystyle y_{0}\in {\mathcal {R}}\left(x_{0}\right)} ). Then for every neighborhood U {\displaystyle U} of x 0 {\displaystyle x_{0}} in X , {\displaystyle X,} y 0 {\displaystyle y_{0}} belongs to the relative interior of R ( U ) {\displaystyle {\mathcal {R}}(U)} in aff ( Im R ) {\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})} (that is, y 0 ∈ int aff ( Im R ) R ( U ) {\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)} ). In particular, if i b ( Im R ) ≠ ∅ {\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing } then i b ( Im R ) = i ( Im R ) = rint ( Im R ) . {\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).} </p> <h2 id="corollaries">Corollaries</h2> <h3>Closed graph theorem</h3> <p><a href="/facts/Closed_graph_theorem/CK2ra00W">Closed graph theorem</a>—Let X {\displaystyle X} and Y {\displaystyle Y} be <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a> and T : X → Y {\displaystyle T:X\to Y} be a linear map. Then T {\displaystyle T} is continuous if and only if the graph of T {\displaystyle T} is closed in X × Y . {\displaystyle X\times Y.} </p> Proof <p>For the non-trivial direction, assume that the graph of T {\displaystyle T} is closed and let R := T − 1 : Y ⇉ X . {\displaystyle {\mathcal {R}}:=T^{-1}:Y\rightrightarrows X.} It is easy to see that gr R {\displaystyle \operatorname {gr} {\mathcal {R}}} is closed and convex and that its image is X . {\displaystyle X.} Given x ∈ X , {\displaystyle x\in X,} ( T x , x ) {\displaystyle (Tx,x)} belongs to Y × X {\displaystyle Y\times X} so that for every open neighborhood V {\displaystyle V} of T x {\displaystyle Tx} in Y , {\displaystyle Y,} R ( V ) = T − 1 ( V ) {\displaystyle {\mathcal {R}}(V)=T^{-1}(V)} is a neighborhood of x {\displaystyle x} in X . {\displaystyle X.} Thus T {\displaystyle T} is continuous at x . {\displaystyle x.} <a href="/facts/Q.E.D./B8I8jPwT">Q.E.D.</a> </p> <h3>Uniform boundedness principle</h3> <p><a href="/facts/Uniform_boundedness_principle/rbbdebsr">Uniform boundedness principle</a>—Let X {\displaystyle X} and Y {\displaystyle Y} be <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a> and T : X → Y {\displaystyle T:X\to Y} be a bijective linear map. Then T {\displaystyle T} is continuous if and only if T − 1 : Y → X {\displaystyle T^{-1}:Y\to X} is continuous. Furthermore, if T {\displaystyle T} is continuous then T {\displaystyle T} is an isomorphism of <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a>. </p> Proof <p>Apply the closed graph theorem to T {\displaystyle T} and T − 1 . {\displaystyle T^{-1}.} Q.E.D. </p> <h3>Open mapping theorem</h3> <p><a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">Open mapping theorem</a>—Let X {\displaystyle X} and Y {\displaystyle Y} be <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a> and T : X → Y {\displaystyle T:X\to Y} be a continuous surjective linear map. Then T is an <a href="/facts/Open_map/abHUKX7l">open map</a>. </p> Proof <p>Clearly, T {\displaystyle T} is a closed and convex relation whose image is Y . {\displaystyle Y.} Let U {\displaystyle U} be a non-empty open subset of X , {\displaystyle X,} let y {\displaystyle y} be in T ( U ) , {\displaystyle T(U),} and let x {\displaystyle x} in U {\displaystyle U} be such that y = T x . {\displaystyle y=Tx.} From the Ursescu theorem it follows that T ( U ) {\displaystyle T(U)} is a neighborhood of y . {\displaystyle y.} Q.E.D. </p> <h3>Additional corollaries</h3> <p>The following notation and notions are used for these corollaries, where R : X ⇉ Y {\displaystyle {\mathcal {R}}:X\rightrightarrows Y} is a set-valued function, S {\displaystyle S} is a non-empty subset of a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> X {\displaystyle X} : </p> <ul><li>a <a href="/facts/Convex_series/0fJ9k937">convex series</a> with elements of S {\displaystyle S} is a <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a> of the form ∑ i = 1 ∞ r i s i {\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}} where all s i ∈ S {\displaystyle s_{i}\in S} and ∑ i = 1 ∞ r i = 1 {\textstyle \sum _{i=1}^{\infty }r_{i}=1} is a series of non-negative numbers. If ∑ i = 1 ∞ r i s i {\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}} converges then the series is called convergent while if ( s i ) i = 1 ∞ {\displaystyle \left(s_{i}\right)_{i=1}^{\infty }} is bounded then the series is called bounded and b-convex.</li> <li> S {\displaystyle S} is ideally convex if any convergent b-convex series of elements of S {\displaystyle S} has its sum in S . {\displaystyle S.} </li> <li> S {\displaystyle S} is lower ideally convex if there exists a <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> Y {\displaystyle Y} such that S {\displaystyle S} is equal to the projection onto X {\displaystyle X} of some ideally convex subset <i>B</i> of X × Y . {\displaystyle X\times Y.} Every ideally convex set is lower ideally convex.</li></ul> <p>Corollary—Let X {\displaystyle X} be a barreled <a href="/facts/First_countable_space/HDuTYa6A">first countable space</a> and let C {\displaystyle C} be a subset of X . {\displaystyle X.} Then: </p> <ol><li>If C {\displaystyle C} is lower ideally convex then C i = int C . {\displaystyle C^{i}=\operatorname {int} C.} </li> <li>If C {\displaystyle C} is ideally convex then C i = int C = int ( cl C ) = ( cl C ) i . {\displaystyle C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.} </li></ol> <h2 id="related-theorems">Related theorems</h2> <h3>Simons' theorem</h3> <p><a href="/facts/Jim_Simons/op947h9w">Simons'</a> theorem<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>—Let X {\displaystyle X} and Y {\displaystyle Y} be <a href="/facts/First_countable/HDuTYa6A">first countable</a> with X {\displaystyle X} locally convex. Suppose that R : X ⇉ Y {\displaystyle {\mathcal {R}}:X\rightrightarrows Y} is a multimap with non-empty domain that satisfies <a href="/facts/Convex_series/0fJ9k937">condition (Hw<i>x</i>)</a> or else assume that X {\displaystyle X} is a <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> and that R {\displaystyle {\mathcal {R}}} is <a href="/facts/Convex_series/0fJ9k937">lower ideally convex</a>. Assume that span ( Im R − y ) {\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)} is <a href="/facts/Barrelled_space/r2H58TN3">barreled</a> for some/every y ∈ Im R . {\displaystyle y\in \operatorname {Im} {\mathcal {R}}.} Assume that y 0 ∈ i ( Im R ) {\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})} and let x 0 ∈ R − 1 ( y 0 ) . {\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).} Then for every neighborhood U {\displaystyle U} of x 0 {\displaystyle x_{0}} in X , {\displaystyle X,} y 0 {\displaystyle y_{0}} belongs to the relative interior of R ( U ) {\displaystyle {\mathcal {R}}(U)} in aff ( Im R ) {\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})} (i.e. y 0 ∈ int aff ( Im R ) R ( U ) {\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)} ). In particular, if i b ( Im R ) ≠ ∅ {\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing } then i b ( Im R ) = i ( Im R ) = rint ( Im R ) . {\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).} </p> <h3>Robinson–Ursescu theorem</h3> <p>The implication (1) ⟹ {\displaystyle \implies } (2) in the following theorem is known as the Robinson–Ursescu theorem.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <p>Robinson–Ursescu theorem<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>—Let ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\,\cdot \,\|)} and ( Y , ‖ ⋅ ‖ ) {\displaystyle (Y,\|\,\cdot \,\|)} be <a href="/facts/Normed_space/rmDljfyE">normed spaces</a> and R : X ⇉ Y {\displaystyle {\mathcal {R}}:X\rightrightarrows Y} be a multimap with non-empty domain. Suppose that Y {\displaystyle Y} is a <a href="/facts/Barreled_space/r2H58TN3">barreled space</a>, the graph of R {\displaystyle {\mathcal {R}}} verifies condition <a href="/facts/Convex_series/0fJ9k937">condition (Hw<i>x</i>)</a>, and that ( x 0 , y 0 ) ∈ gr R . {\displaystyle (x_{0},y_{0})\in \operatorname {gr} {\mathcal {R}}.} Let C X {\displaystyle C_{X}} (resp. C Y {\displaystyle C_{Y}} ) denote the closed unit ball in X {\displaystyle X} (resp. Y {\displaystyle Y} ) (so C X = { x ∈ X : ‖ x ‖ ≤ 1 } {\displaystyle C_{X}=\{x\in X:\|x\|\leq 1\}} ). Then the following are equivalent: </p> <ol><li> y 0 {\displaystyle y_{0}} belongs to the <a href="/facts/Algebraic_interior/oCWBqUr9">algebraic interior</a> of Im R . {\displaystyle \operatorname {Im} {\mathcal {R}}.} </li> <li> y 0 ∈ int R ( x 0 + C X ) . {\displaystyle y_{0}\in \operatorname {int} {\mathcal {R}}\left(x_{0}+C_{X}\right).} </li> <li>There exists B > 0 {\displaystyle B>0} such that for all 0 ≤ r ≤ 1 , {\displaystyle 0\leq r\leq 1,} y 0 + B r C Y ⊆ R ( x 0 + r C X ) . {\displaystyle y_{0}+BrC_{Y}\subseteq {\mathcal {R}}\left(x_{0}+rC_{X}\right).} </li> <li>There exist A > 0 {\displaystyle A>0} and B > 0 {\displaystyle B>0} such that for all x ∈ x 0 + A C X {\displaystyle x\in x_{0}+AC_{X}} and all y ∈ y 0 + A C Y , {\displaystyle y\in y_{0}+AC_{Y},} d ( x , R − 1 ( y ) ) ≤ B ⋅ d ( y , R ( x ) ) . {\displaystyle d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq B\cdot d(y,{\mathcal {R}}(x)).} </li> <li>There exists B > 0 {\displaystyle B>0} such that for all x ∈ X {\displaystyle x\in X} and all y ∈ y 0 + B C Y , {\displaystyle y\in y_{0}+BC_{Y},} d ( x , R − 1 ( y ) ) ≤ 1 + ‖ x − x 0 ‖ B − ‖ y − y 0 ‖ ⋅ d ( y , R ( x ) ) . {\displaystyle d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq {\frac {1+\left\|x-x_{0}\right\|}{B-\left\|y-y_{0}\right\|}}\cdot d(y,{\mathcal {R}}(x)).} </li></ol> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Closed_graph_theorem/CK2ra00W">Closed graph theorem</a> – Theorem relating continuity to graphs</li> <li><a href="/facts/Closed_graph_theorem_(functional_analysis)/l01mnNoJ">Closed graph theorem (functional analysis)</a> – Theorems connecting continuity to closure of graphs</li> <li><a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">Open mapping theorem (functional analysis)</a> – Condition for a linear operator to be open</li> <li><a href="/facts/Surjection_of_Fr%25C3%25A9chet_spaces/06MelJYo">Surjection of Fréchet spaces</a> – Characterization of surjectivity</li> <li><a href="/facts/Uniform_boundedness_principle/rbbdebsr">Uniform boundedness principle</a> – Theorem stating that pointwise boundedness implies uniform boundedness</li> <li><a href="/facts/Webbed_space/ChtDnm0X">Webbed space</a> – Space where open mapping and closed graph theorems hold</li></ul> <h2 id="notes">Notes</h2> <ul><li>Zălinescu, Constantin (30 July 2002). <a href="https://archive.org/details/convexanalysisge00zali_934"><i>Convex Analysis in General Vector Spaces</i></a>. River Edge, N.J. London: <a href="/facts/World_Scientific_Publishing/qJPTP0Yk">World Scientific Publishing</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-4488-15-0. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1921556">1921556</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/285163112">285163112</a> – via <a href="/facts/Internet_Archive/PHStU8Lo">Internet Archive</a>.</li> <li>Baggs, Ivan (1974). <a href="https://www.ams.org/">"Functions with a closed graph"</a>. <i>Proceedings of the American Mathematical Society</i>. 43 (2): 439–442. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9939-1974-0334132-8">10.1090/S0002-9939-1974-0334132-8</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9939">0002-9939</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Zălinescu 2002, p. 23. - 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. <a href="https://archive.org/details/convexanalysisge00zali_934" target="_blank">https://archive.org/details/convexanalysisge00zali_934</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Zălinescu 2002, p. 22-23. - 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. <a href="https://archive.org/details/convexanalysisge00zali_934" target="_blank">https://archive.org/details/convexanalysisge00zali_934</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Zălinescu 2002, p. 24. - 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. <a href="https://archive.org/details/convexanalysisge00zali_934" target="_blank">https://archive.org/details/convexanalysisge00zali_934</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Zălinescu 2002, p. 24. - 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. <a href="https://archive.org/details/convexanalysisge00zali_934" target="_blank">https://archive.org/details/convexanalysisge00zali_934</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>