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Surjection of Fréchet spaces
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<h2 id="preliminaries-definitions-and-notation">Preliminaries, definitions, and notation</h2> <p>Let L : X → Y {\displaystyle L:X\to Y} be a continuous linear map between <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a>. </p><p>The <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a> of X {\displaystyle X} is denoted by X ′ . {\displaystyle X^{\prime }.} </p><p>The <a href="/facts/Transpose_of_a_linear_map/uv0JWTHq">transpose</a> of L {\displaystyle L} is the map t L : Y ′ → X ′ {\displaystyle {}^{t}L:Y^{\prime }\to X^{\prime }} defined by L ( y ′ ) := y ′ ∘ L . {\displaystyle L\left(y^{\prime }\right):=y^{\prime }\circ L.} If L : X → Y {\displaystyle L:X\to Y} is surjective then t L : Y ′ → X ′ {\displaystyle {}^{t}L:Y^{\prime }\to X^{\prime }} will be <a href="/facts/Injective/5ES6tqf5">injective</a>, but the converse is not true in general. </p><p>The <a href="/facts/Weak_topology/XJASE9Up">weak topology</a> on X {\displaystyle X} (resp. X ′ {\displaystyle X^{\prime }} ) is denoted by σ ( X , X ′ ) {\displaystyle \sigma \left(X,X^{\prime }\right)} (resp. σ ( X ′ , X ) {\displaystyle \sigma \left(X^{\prime },X\right)} ). The set X {\displaystyle X} endowed with this topology is denoted by ( X , σ ( X , X ′ ) ) . {\displaystyle \left(X,\sigma \left(X,X^{\prime }\right)\right).} The topology σ ( X , X ′ ) {\displaystyle \sigma \left(X,X^{\prime }\right)} is the weakest topology on X {\displaystyle X} making all linear functionals in X ′ {\displaystyle X^{\prime }} continuous. </p><p>If S ⊆ Y {\displaystyle S\subseteq Y} then the <a href="/facts/Polar_set/JY45pfbt">polar</a> of S {\displaystyle S} in Y {\displaystyle Y} is denoted by S ∘ . {\displaystyle S^{\circ }.} </p><p>If p : X → R {\displaystyle p:X\to \mathbb {R} } is a <a href="/facts/Seminorm/vMuescKp">seminorm</a> on X {\displaystyle X} , then X p {\displaystyle X_{p}} will denoted the vector space X {\displaystyle X} endowed with the weakest <a href="/facts/Topological_vector_space/XRex1ChN">TVS</a> topology making p {\displaystyle p} continuous.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> A neighborhood basis of X p {\displaystyle X_{p}} at the origin consists of the sets { x ∈ X : p ( x ) < r } {\displaystyle \left\{x\in X:p(x)<r\right\}} as r {\displaystyle r} ranges over the positive reals. If p {\displaystyle p} is not a norm then X p {\displaystyle X_{p}} is not <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> and ker p := { x ∈ X : p ( x ) = 0 } {\displaystyle \ker p:=\left\{x\in X:p(x)=0\right\}} is a linear subspace of X {\displaystyle X} . If p {\displaystyle p} is continuous then the identity map Id : X → X p {\displaystyle \operatorname {Id} :X\to X_{p}} is continuous so we may identify the continuous dual space X p ′ {\displaystyle X_{p}^{\prime }} of X p {\displaystyle X_{p}} as a subset of X ′ {\displaystyle X^{\prime }} via the transpose of the identity map t Id : X p ′ → X ′ , {\displaystyle {}^{t}\operatorname {Id} :X_{p}^{\prime }\to X^{\prime },} which is <a href="/facts/Injective_function/5ES6tqf5">injective</a>. </p> <h2 id="surjection-of-frchet-spaces">Surjection of Fréchet spaces</h2> <p>Theorem<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> (Banach)—If L : X → Y {\displaystyle L:X\to Y} is a continuous linear map between two Fréchet spaces, then L : X → Y {\displaystyle L:X\to Y} is surjective if and only if the following two conditions both hold: </p> <ol> <li> t L : Y ′ → X ′ {\displaystyle {}^{t}L:Y^{\prime }\to X^{\prime }} is <a href="/facts/Injective/5ES6tqf5">injective</a>, and</li> <li>the <a href="/facts/Image_of_a_function/KANefMAl">image</a> of t L , {\displaystyle {}^{t}L,} denoted by Im t L , {\displaystyle \operatorname {Im} {}^{t}L,} is weakly closed in X ′ {\displaystyle X^{\prime }} (i.e. closed when X ′ {\displaystyle X^{\prime }} is endowed with the weak-* topology). </li> </ol> <h2 id="extensions-of-the-theorem">Extensions of the theorem</h2> <p>Theorem<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>—If L : X → Y {\displaystyle L:X\to Y} is a continuous linear map between two Fréchet spaces then the following are equivalent: </p> <ol> <li> L : X → Y {\displaystyle L:X\to Y} is surjective. </li> <li>The following two conditions hold: <ol> <li> t L : Y ′ → X ′ {\displaystyle {}^{t}L:Y^{\prime }\to X^{\prime }} is <a href="/facts/Injective/5ES6tqf5">injective</a>;</li> <li>the <a href="/facts/Image_of_a_function/KANefMAl">image</a> Im t L {\displaystyle \operatorname {Im} {}^{t}L} of t L {\displaystyle {}^{t}L} is weakly closed in X ′ . {\displaystyle X^{\prime }.} </li> </ol> </li> <li>For every continuous seminorm p {\displaystyle p} on X {\displaystyle X} there exists a continuous seminorm q {\displaystyle q} on Y {\displaystyle Y} such that the following are true: <ol> <li>for every y ∈ Y {\displaystyle y\in Y} there exists some x ∈ X {\displaystyle x\in X} such that q ( L ( x ) − y ) = 0 {\displaystyle q(L(x)-y)=0} ;</li> <li>for every y ′ ∈ Y , {\displaystyle y^{\prime }\in Y,} if t L ( y ′ ) ∈ X p ′ {\displaystyle {}^{t}L\left(y^{\prime }\right)\in X_{p}^{\prime }} then y ′ ∈ Y q ′ . {\displaystyle y^{\prime }\in Y_{q}^{\prime }.} </li> </ol> </li> <li>For every continuous seminorm p {\displaystyle p} on X {\displaystyle X} there exists a linear subspace N {\displaystyle N} of Y {\displaystyle Y} such that the following are true: <ol> <li>for every y ∈ Y {\displaystyle y\in Y} there exists some x ∈ X {\displaystyle x\in X} such that L ( x ) − y ∈ N {\displaystyle L(x)-y\in N} ;</li> <li>for every y ′ ∈ Y ′ , {\displaystyle y^{\prime }\in Y^{\prime },} if t L ( y ′ ) ∈ X p ′ {\displaystyle {}^{t}L\left(y^{\prime }\right)\in X_{p}^{\prime }} then y ′ ∈ N ∘ . {\displaystyle y^{\prime }\in N^{\circ }.} </li> </ol> </li> <li>There is a <a href="/facts/Non-increasing/gV2S4uqp">non-increasing</a> sequence N 1 ⊇ N 2 ⊇ N 3 ⊇ ⋯ {\displaystyle N_{1}\supseteq N_{2}\supseteq N_{3}\supseteq \cdots } of closed linear subspaces of Y {\displaystyle Y} whose intersection is equal to { 0 } {\displaystyle \{0\}} and such that the following are true: <ol> <li>for every y ∈ Y {\displaystyle y\in Y} and every positive integer k {\displaystyle k} , there exists some x ∈ X {\displaystyle x\in X} such that L ( x ) − y ∈ N k {\displaystyle L(x)-y\in N_{k}} ;</li> <li>for every continuous seminorm p {\displaystyle p} on X {\displaystyle X} there exists an integer k {\displaystyle k} such that any x ∈ X {\displaystyle x\in X} that satisfies L ( x ) ∈ N k {\displaystyle L(x)\in N_{k}} is the limit, in the sense of the seminorm p {\displaystyle p} , of a sequence x 1 , x 2 , … {\displaystyle x_{1},x_{2},\ldots } in elements of X {\displaystyle X} such that L ( x i ) = 0 {\displaystyle L\left(x_{i}\right)=0} for all i . {\displaystyle i.} </li> </ol> </li> </ol> <h2 id="lemmas">Lemmas</h2> <p>The following lemmas are used to prove the theorems on the surjectivity of Fréchet spaces. They are useful even on their own. </p> <p>Theorem<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>—Let X {\displaystyle X} be a Fréchet space and Z {\displaystyle Z} be a linear subspace of X ′ . {\displaystyle X^{\prime }.} The following are equivalent: </p> <ol> <li> Z {\displaystyle Z} is weakly closed in X ′ {\displaystyle X^{\prime }} ;</li> <li>There exists a basis B {\displaystyle {\mathcal {B}}} of neighborhoods of the origin of X {\displaystyle X} such that for every B ∈ B , {\displaystyle B\in {\mathcal {B}},} B ∘ ∩ Z {\displaystyle B^{\circ }\cap Z} is weakly closed;</li> <li>The intersection of Z {\displaystyle Z} with every equicontinuous subset E {\displaystyle E} of X ′ {\displaystyle X^{\prime }} is relatively closed in E {\displaystyle E} (where X ′ {\displaystyle X^{\prime }} is given the weak topology induced by X {\displaystyle X} and E {\displaystyle E} is given the subspace topology induced by X ′ {\displaystyle X^{\prime }} ).</li> </ol> <p>Theorem<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>—On the dual X ′ {\displaystyle X^{\prime }} of a Fréchet space X {\displaystyle X} , the topology of uniform convergence on compact convex subsets of X {\displaystyle X} is identical to the topology of <a href="/facts/Uniform_convergence_on_compact_subsets/65F9goFE">uniform convergence on compact subsets</a> of X {\displaystyle X} . </p> <p>Theorem<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>—Let L : X → Y {\displaystyle L:X\to Y} be a linear map between Hausdorff <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> TVSs, with X {\displaystyle X} also <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a>. If the map L : ( X , σ ( X , X ′ ) ) → ( Y , σ ( Y , Y ′ ) ) {\displaystyle L:\left(X,\sigma \left(X,X^{\prime }\right)\right)\to \left(Y,\sigma \left(Y,Y^{\prime }\right)\right)} is continuous then L : X → Y {\displaystyle L:X\to Y} is continuous (where X {\displaystyle X} and Y {\displaystyle Y} carry their original topologies). </p> <h2 id="applications">Applications</h2> <h3>Borel's theorem on power series expansions</h3> <p>Theorem<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> (E. Borel)—Fix a positive integer n {\displaystyle n} . If P {\displaystyle P} is an arbitrary <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> in n {\displaystyle n} indeterminates with complex coefficients then there exists a C ∞ {\displaystyle {\mathcal {C}}^{\infty }} function f : R n → C {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {C} } whose Taylor expansion at the origin is identical to P {\displaystyle P} . </p><p>That is, suppose that for every n {\displaystyle n} -tuple of non-negative integers p = ( p 1 , … , p n ) {\displaystyle p=\left(p_{1},\ldots ,p_{n}\right)} we are given a complex number a p {\displaystyle a_{p}} (with no restrictions). Then there exists a C ∞ {\displaystyle {\mathcal {C}}^{\infty }} function f : R n → C {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {C} } such that a p = ( ∂ / ∂ x ) p f | x = 0 {\displaystyle a_{p}=\left(\partial /\partial x\right)^{p}f{\bigg \vert }_{x=0}} for every n {\displaystyle n} -tuple p . {\displaystyle p.} </p> <h3>Linear partial differential operators</h3> <p class="note">See also: <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">Distribution (mathematics)</a></p> <p>Theorem<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>—Let D {\displaystyle D} be a linear partial differential operator with C ∞ {\displaystyle {\mathcal {C}}^{\infty }} coefficients in an open subset U ⊆ R n . {\displaystyle U\subseteq \mathbb {R} ^{n}.} The following are equivalent: </p> <ol> <li>For every f ∈ C ∞ ( U ) {\displaystyle f\in {\mathcal {C}}^{\infty }(U)} there exists some u ∈ C ∞ ( U ) {\displaystyle u\in {\mathcal {C}}^{\infty }(U)} such that D u = f . {\displaystyle Du=f.} </li> <li> U {\displaystyle U} is D {\displaystyle D} -convex and D {\displaystyle D} is semiglobally solvable.</li> </ol> <p> D {\displaystyle D} being semiglobally solvable in U {\displaystyle U} means that for every <a href="/facts/Relatively_compact/z5uIqMDp">relatively compact</a> open subset V {\displaystyle V} of U {\displaystyle U} , the following condition holds: </p> to every f ∈ C ∞ ( U ) {\displaystyle f\in {\mathcal {C}}^{\infty }(U)} there is some g ∈ C ∞ ( U ) {\displaystyle g\in {\mathcal {C}}^{\infty }(U)} such that D g = f {\displaystyle Dg=f} in V {\displaystyle V} . <p> U {\displaystyle U} being D {\displaystyle D} -convex means that for every compact subset K ⊆ U {\displaystyle K\subseteq U} and every integer n ≥ 0 , {\displaystyle n\geq 0,} there is a compact subset C n {\displaystyle C_{n}} of U {\displaystyle U} such that for every <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distribution</a> d {\displaystyle d} with compact support in U {\displaystyle U} , the following condition holds: </p> if t D d {\displaystyle {}^{t}Dd} is of order ≤ n {\displaystyle \leq n} and if supp t D d ⊆ K , {\displaystyle \operatorname {supp} {}^{t}Dd\subseteq K,} then supp d ⊆ C n . {\displaystyle \operatorname {supp} d\subseteq C_{n}.} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Epimorphism/IrzonI01">Epimorphism</a> – Surjective homomorphism</li> <li><a href="/facts/Exponential_formula/tF9K4Rj9">Exponential formula</a></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></ul> <h2 id="bibliography">Bibliography</h2> <ul><li>Narici, Lawrence; Beckenstein, Edward (2011). <i>Topological Vector Spaces</i>. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li> <li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li> <li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Trèves 2006, pp. 378–384. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Trèves 2006, pp. 378–384. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Trèves 2006, pp. 378–384. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Trèves 2006, pp. 378–384. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Trèves 2006, pp. 378–384. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Trèves 2006, pp. 378–384. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Trèves 2006, pp. 378–384. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Trèves 2006, p. 390. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Trèves 2006, p. 392. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>