Norm derivatives and characterizations of inner product spaces /:
The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordan-von Neumann's classical theorem...
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | , |
| Format: | Electronic eBook |
| Language: | English |
| Published: |
New Jersey :
World Scientific,
©2010.
|
| Subjects: | |
| Online Access: | DE-862 DE-863 |
| Summary: | The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordan-von Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces. This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces. Thus the book can serve as an advanced undergraduate or graduate text as well as a resource book for researchers working in geometry of Banach (Hilbert) spaces or in the theory of functional equations (and their applications). |
| Physical Description: | 1 online resource (x, 188 pages) : illustrations |
| Bibliography: | Includes bibliographical references (pages 179-185) and index. |
| ISBN: | 9789814287272 981428727X |
Staff View
MARC
| LEADER | 00000cam a2200000 a 4500 | ||
|---|---|---|---|
| 001 | ZDB-4-EBA-ocn630153547 | ||
| 003 | OCoLC | ||
| 005 | 20250103110447.0 | ||
| 006 | m o d | ||
| 007 | cr muu|||uu||| | ||
| 008 | 100524s2010 njua ob 001 0 eng d | ||
| 040 | |a LLB |b eng |e pn |c LLB |d OCLCQ |d N$T |d E7B |d YDXCP |d OSU |d OCLCQ |d DEBSZ |d OCLCQ |d OCLCF |d OCLCQ |d AGLDB |d ZCU |d MERUC |d OCLCQ |d U3W |d OCLCQ |d VTS |d ICG |d INT |d OCLCQ |d STF |d DKC |d AU@ |d OCLCQ |d M8D |d OCLCO |d OCLCQ |d OCLCO |d OCLCQ |d OCLCL | ||
| 019 | |a 670430145 |a 694144276 | ||
| 020 | |a 9789814287272 |q (electronic bk.) | ||
| 020 | |a 981428727X |q (electronic bk.) | ||
| 020 | |z 9789814287265 | ||
| 020 | |z 9814287261 | ||
| 035 | |a (OCoLC)630153547 |z (OCoLC)670430145 |z (OCoLC)694144276 | ||
| 050 | 4 | |a QA322.4 |b A57 2010eb | |
| 072 | 7 | |a MAT |x 031000 |2 bisacsh | |
| 082 | 7 | |a 515.733 |2 22 | |
| 049 | |a MAIN | ||
| 100 | 1 | |a Alsina, Claudi. |0 http://id.loc.gov/authorities/names/n79030525 | |
| 245 | 1 | 0 | |a Norm derivatives and characterizations of inner product spaces / |c Claudi Alsina, Justyna Sikorska, M Santos Tomás. |
| 260 | |a New Jersey : |b World Scientific, |c ©2010. | ||
| 300 | |a 1 online resource (x, 188 pages) : |b illustrations | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 520 | |a The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordan-von Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces. This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces. Thus the book can serve as an advanced undergraduate or graduate text as well as a resource book for researchers working in geometry of Banach (Hilbert) spaces or in the theory of functional equations (and their applications). | ||
| 504 | |a Includes bibliographical references (pages 179-185) and index. | ||
| 505 | 0 | |a Introduction. Historical notes -- Normed linear spaces -- Strictly convex normed linear spaces -- Inner product spaces --Orthogonalities in normed linear spaces -- Norm derivatives. Norm derivatives : definition and basic properties -- Orthogonality relations based on norm derivatives -- p'[symbol]-orthogonal transformations -- On the equivalence of two norm derivatives -- Norm derivatives and projections in normed linear spaces -- Norm derivatives and Lagrange's identity in normed linear spaces -- On some extensions of the norm derivatives -- p-orthogonal additivity -- Norm derivatives and heights. Definition and basic properties -- Characterizations of inner product spaces involving geometrical properties of a height in a triangle -- Height functions and classical orthogonalities -- A new orthogonality relation -- Orthocenters -- A characterization of inner product spaces involving an isosceles trapezoid property -- Functional equations of the height transform -- Perpendicular bisectors in Normed spaces. Definitions and basic properties -- A new orthogonality relation -- Relations between perpendicular bisectors and classical orthogonalities -- On the radius of the circumscribed circumference of a triangle -- Circumcenters in a triangle -- Euler line in real normed space -- Functional equation of the perpendicular bisector transform -- Bisectrices in real Normed spaces. Bisectrices in real normed spaces -- A new orthogonality relation -- Functional equation of the bisectrix transform -- Generalized bisectrices in strictly convex real normed spaces -- Incenters and generalized bisectrices -- Areas of triangles in Normed spaces. Definition of four areas of triangles -- Classical properties of the areas and characterizations of inner product spaces -- Equalities between different area functions -- The area orthogonality. | |
| 588 | 0 | |a Print version record. | |
| 650 | 0 | |a Normed linear spaces. |0 http://id.loc.gov/authorities/subjects/sh85092434 | |
| 650 | 0 | |a Inner product spaces. |0 http://id.loc.gov/authorities/subjects/sh85066514 | |
| 650 | 6 | |a Espaces linéaires normés. | |
| 650 | 6 | |a Espaces à produit scalaire. | |
| 650 | 7 | |a MATHEMATICS |x Transformations. |2 bisacsh | |
| 650 | 7 | |a Inner product spaces |2 fast | |
| 650 | 7 | |a Normed linear spaces |2 fast | |
| 655 | 4 | |a Electronic book. | |
| 700 | 1 | |a Sikorska, Justyna. |0 http://id.loc.gov/authorities/names/no2010059471 | |
| 700 | 1 | |a Tomás, M. Santos |q (Maria Santos) |1 https://id.oclc.org/worldcat/entity/E39PBJrwTYjJFkDtJypFQ3CPcP |0 http://id.loc.gov/authorities/names/no2010061277 | |
| 758 | |i has work: |a Norm derivatives and characterizations of inner product spaces (Text) |1 https://id.oclc.org/worldcat/entity/E39PCGBFhDPDcCHwycY4VRJhpP |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Alsina, Claudi. |t Norm derivatives and characterizations of inner product spaces. |d Singapore : World Scientific, ©2010 |z 9789814287265 |w (OCoLC)540206466 |
| 966 | 4 | 0 | |l DE-862 |p ZDB-4-EBA |q FWS_PDA_EBA |u https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=340640 |3 Volltext |
| 966 | 4 | 0 | |l DE-863 |p ZDB-4-EBA |q FWS_PDA_EBA |u https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=340640 |3 Volltext |
| 938 | |a ebrary |b EBRY |n ebr10422427 | ||
| 938 | |a EBSCOhost |b EBSC |n 340640 | ||
| 938 | |a YBP Library Services |b YANK |n 3511491 | ||
| 994 | |a 92 |b GEBAY | ||
| 912 | |a ZDB-4-EBA | ||
| 049 | |a DE-862 | ||
| 049 | |a DE-863 | ||
Record in the Search Index
| DE-BY-FWS_katkey | ZDB-4-EBA-ocn630153547 |
|---|---|
| _version_ | 1829094664989310976 |
| adam_text | |
| any_adam_object | |
| author | Alsina, Claudi |
| author2 | Sikorska, Justyna Tomás, M. Santos (Maria Santos) |
| author2_role | |
| author2_variant | j s js m s t ms mst |
| author_GND | http://id.loc.gov/authorities/names/n79030525 http://id.loc.gov/authorities/names/no2010059471 http://id.loc.gov/authorities/names/no2010061277 |
| author_facet | Alsina, Claudi Sikorska, Justyna Tomás, M. Santos (Maria Santos) |
| author_role | |
| author_sort | Alsina, Claudi |
| author_variant | c a ca |
| building | Verbundindex |
| bvnumber | localFWS |
| callnumber-first | Q - Science |
| callnumber-label | QA322 |
| callnumber-raw | QA322.4 A57 2010eb |
| callnumber-search | QA322.4 A57 2010eb |
| callnumber-sort | QA 3322.4 A57 42010EB |
| callnumber-subject | QA - Mathematics |
| collection | ZDB-4-EBA |
| contents | Introduction. Historical notes -- Normed linear spaces -- Strictly convex normed linear spaces -- Inner product spaces --Orthogonalities in normed linear spaces -- Norm derivatives. Norm derivatives : definition and basic properties -- Orthogonality relations based on norm derivatives -- p'[symbol]-orthogonal transformations -- On the equivalence of two norm derivatives -- Norm derivatives and projections in normed linear spaces -- Norm derivatives and Lagrange's identity in normed linear spaces -- On some extensions of the norm derivatives -- p-orthogonal additivity -- Norm derivatives and heights. Definition and basic properties -- Characterizations of inner product spaces involving geometrical properties of a height in a triangle -- Height functions and classical orthogonalities -- A new orthogonality relation -- Orthocenters -- A characterization of inner product spaces involving an isosceles trapezoid property -- Functional equations of the height transform -- Perpendicular bisectors in Normed spaces. Definitions and basic properties -- A new orthogonality relation -- Relations between perpendicular bisectors and classical orthogonalities -- On the radius of the circumscribed circumference of a triangle -- Circumcenters in a triangle -- Euler line in real normed space -- Functional equation of the perpendicular bisector transform -- Bisectrices in real Normed spaces. Bisectrices in real normed spaces -- A new orthogonality relation -- Functional equation of the bisectrix transform -- Generalized bisectrices in strictly convex real normed spaces -- Incenters and generalized bisectrices -- Areas of triangles in Normed spaces. Definition of four areas of triangles -- Classical properties of the areas and characterizations of inner product spaces -- Equalities between different area functions -- The area orthogonality. |
| ctrlnum | (OCoLC)630153547 |
| dewey-full | 515.733 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.733 |
| dewey-search | 515.733 |
| dewey-sort | 3515.733 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>05831cam a2200565 a 4500</leader><controlfield tag="001">ZDB-4-EBA-ocn630153547</controlfield><controlfield tag="003">OCoLC</controlfield><controlfield tag="005">20250103110447.0</controlfield><controlfield tag="006">m o d </controlfield><controlfield tag="007">cr muu|||uu|||</controlfield><controlfield tag="008">100524s2010 njua ob 001 0 eng d</controlfield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">LLB</subfield><subfield code="b">eng</subfield><subfield code="e">pn</subfield><subfield code="c">LLB</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">N$T</subfield><subfield code="d">E7B</subfield><subfield code="d">YDXCP</subfield><subfield code="d">OSU</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">DEBSZ</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">OCLCF</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">AGLDB</subfield><subfield code="d">ZCU</subfield><subfield code="d">MERUC</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">U3W</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">VTS</subfield><subfield code="d">ICG</subfield><subfield code="d">INT</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">STF</subfield><subfield code="d">DKC</subfield><subfield code="d">AU@</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">M8D</subfield><subfield code="d">OCLCO</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">OCLCO</subfield><subfield code="d">OCLCQ</subfield><subfield code="d">OCLCL</subfield></datafield><datafield tag="019" ind1=" " ind2=" "><subfield code="a">670430145</subfield><subfield code="a">694144276</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9789814287272</subfield><subfield code="q">(electronic bk.)</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">981428727X</subfield><subfield code="q">(electronic bk.)</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="z">9789814287265</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="z">9814287261</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)630153547</subfield><subfield code="z">(OCoLC)670430145</subfield><subfield code="z">(OCoLC)694144276</subfield></datafield><datafield tag="050" ind1=" " ind2="4"><subfield code="a">QA322.4</subfield><subfield code="b">A57 2010eb</subfield></datafield><datafield tag="072" ind1=" " ind2="7"><subfield code="a">MAT</subfield><subfield code="x">031000</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="082" ind1="7" ind2=" "><subfield code="a">515.733</subfield><subfield code="2">22</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">MAIN</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Alsina, Claudi.</subfield><subfield code="0">http://id.loc.gov/authorities/names/n79030525</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Norm derivatives and characterizations of inner product spaces /</subfield><subfield code="c">Claudi Alsina, Justyna Sikorska, M Santos Tomás.</subfield></datafield><datafield tag="260" ind1=" " ind2=" "><subfield code="a">New Jersey :</subfield><subfield code="b">World Scientific,</subfield><subfield code="c">©2010.</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 online resource (x, 188 pages) :</subfield><subfield code="b">illustrations</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">computer</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">online resource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordan-von Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces. This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces. Thus the book can serve as an advanced undergraduate or graduate text as well as a resource book for researchers working in geometry of Banach (Hilbert) spaces or in the theory of functional equations (and their applications).</subfield></datafield><datafield tag="504" ind1=" " ind2=" "><subfield code="a">Includes bibliographical references (pages 179-185) and index.</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">Introduction. Historical notes -- Normed linear spaces -- Strictly convex normed linear spaces -- Inner product spaces --Orthogonalities in normed linear spaces -- Norm derivatives. Norm derivatives : definition and basic properties -- Orthogonality relations based on norm derivatives -- p'[symbol]-orthogonal transformations -- On the equivalence of two norm derivatives -- Norm derivatives and projections in normed linear spaces -- Norm derivatives and Lagrange's identity in normed linear spaces -- On some extensions of the norm derivatives -- p-orthogonal additivity -- Norm derivatives and heights. Definition and basic properties -- Characterizations of inner product spaces involving geometrical properties of a height in a triangle -- Height functions and classical orthogonalities -- A new orthogonality relation -- Orthocenters -- A characterization of inner product spaces involving an isosceles trapezoid property -- Functional equations of the height transform -- Perpendicular bisectors in Normed spaces. Definitions and basic properties -- A new orthogonality relation -- Relations between perpendicular bisectors and classical orthogonalities -- On the radius of the circumscribed circumference of a triangle -- Circumcenters in a triangle -- Euler line in real normed space -- Functional equation of the perpendicular bisector transform -- Bisectrices in real Normed spaces. Bisectrices in real normed spaces -- A new orthogonality relation -- Functional equation of the bisectrix transform -- Generalized bisectrices in strictly convex real normed spaces -- Incenters and generalized bisectrices -- Areas of triangles in Normed spaces. Definition of four areas of triangles -- Classical properties of the areas and characterizations of inner product spaces -- Equalities between different area functions -- The area orthogonality.</subfield></datafield><datafield tag="588" ind1="0" ind2=" "><subfield code="a">Print version record.</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Normed linear spaces.</subfield><subfield code="0">http://id.loc.gov/authorities/subjects/sh85092434</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Inner product spaces.</subfield><subfield code="0">http://id.loc.gov/authorities/subjects/sh85066514</subfield></datafield><datafield tag="650" ind1=" " ind2="6"><subfield code="a">Espaces linéaires normés.</subfield></datafield><datafield tag="650" ind1=" " ind2="6"><subfield code="a">Espaces à produit scalaire.</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">MATHEMATICS</subfield><subfield code="x">Transformations.</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Inner product spaces</subfield><subfield code="2">fast</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Normed linear spaces</subfield><subfield code="2">fast</subfield></datafield><datafield tag="655" ind1=" " ind2="4"><subfield code="a">Electronic book.</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Sikorska, Justyna.</subfield><subfield code="0">http://id.loc.gov/authorities/names/no2010059471</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Tomás, M. Santos</subfield><subfield code="q">(Maria Santos)</subfield><subfield code="1">https://id.oclc.org/worldcat/entity/E39PBJrwTYjJFkDtJypFQ3CPcP</subfield><subfield code="0">http://id.loc.gov/authorities/names/no2010061277</subfield></datafield><datafield tag="758" ind1=" " ind2=" "><subfield code="i">has work:</subfield><subfield code="a">Norm derivatives and characterizations of inner product spaces (Text)</subfield><subfield code="1">https://id.oclc.org/worldcat/entity/E39PCGBFhDPDcCHwycY4VRJhpP</subfield><subfield code="4">https://id.oclc.org/worldcat/ontology/hasWork</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Print version:</subfield><subfield code="a">Alsina, Claudi.</subfield><subfield code="t">Norm derivatives and characterizations of inner product spaces.</subfield><subfield code="d">Singapore : World Scientific, ©2010</subfield><subfield code="z">9789814287265</subfield><subfield code="w">(OCoLC)540206466</subfield></datafield><datafield tag="966" ind1="4" ind2="0"><subfield code="l">DE-862</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FWS_PDA_EBA</subfield><subfield code="u">https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=340640</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="966" ind1="4" ind2="0"><subfield code="l">DE-863</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FWS_PDA_EBA</subfield><subfield code="u">https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=340640</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">ebrary</subfield><subfield code="b">EBRY</subfield><subfield code="n">ebr10422427</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">EBSCOhost</subfield><subfield code="b">EBSC</subfield><subfield code="n">340640</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">YBP Library Services</subfield><subfield code="b">YANK</subfield><subfield code="n">3511491</subfield></datafield><datafield tag="994" ind1=" " ind2=" "><subfield code="a">92</subfield><subfield code="b">GEBAY</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-4-EBA</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-862</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-863</subfield></datafield></record></collection> |
| genre | Electronic book. |
| genre_facet | Electronic book. |
| id | ZDB-4-EBA-ocn630153547 |
| illustrated | Illustrated |
| indexdate | 2025-04-11T08:36:44Z |
| institution | BVB |
| isbn | 9789814287272 981428727X |
| language | English |
| oclc_num | 630153547 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (x, 188 pages) : illustrations |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | World Scientific, |
| record_format | marc |
| spelling | Alsina, Claudi. http://id.loc.gov/authorities/names/n79030525 Norm derivatives and characterizations of inner product spaces / Claudi Alsina, Justyna Sikorska, M Santos Tomás. New Jersey : World Scientific, ©2010. 1 online resource (x, 188 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordan-von Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces. This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces. Thus the book can serve as an advanced undergraduate or graduate text as well as a resource book for researchers working in geometry of Banach (Hilbert) spaces or in the theory of functional equations (and their applications). Includes bibliographical references (pages 179-185) and index. Introduction. Historical notes -- Normed linear spaces -- Strictly convex normed linear spaces -- Inner product spaces --Orthogonalities in normed linear spaces -- Norm derivatives. Norm derivatives : definition and basic properties -- Orthogonality relations based on norm derivatives -- p'[symbol]-orthogonal transformations -- On the equivalence of two norm derivatives -- Norm derivatives and projections in normed linear spaces -- Norm derivatives and Lagrange's identity in normed linear spaces -- On some extensions of the norm derivatives -- p-orthogonal additivity -- Norm derivatives and heights. Definition and basic properties -- Characterizations of inner product spaces involving geometrical properties of a height in a triangle -- Height functions and classical orthogonalities -- A new orthogonality relation -- Orthocenters -- A characterization of inner product spaces involving an isosceles trapezoid property -- Functional equations of the height transform -- Perpendicular bisectors in Normed spaces. Definitions and basic properties -- A new orthogonality relation -- Relations between perpendicular bisectors and classical orthogonalities -- On the radius of the circumscribed circumference of a triangle -- Circumcenters in a triangle -- Euler line in real normed space -- Functional equation of the perpendicular bisector transform -- Bisectrices in real Normed spaces. Bisectrices in real normed spaces -- A new orthogonality relation -- Functional equation of the bisectrix transform -- Generalized bisectrices in strictly convex real normed spaces -- Incenters and generalized bisectrices -- Areas of triangles in Normed spaces. Definition of four areas of triangles -- Classical properties of the areas and characterizations of inner product spaces -- Equalities between different area functions -- The area orthogonality. Print version record. Normed linear spaces. http://id.loc.gov/authorities/subjects/sh85092434 Inner product spaces. http://id.loc.gov/authorities/subjects/sh85066514 Espaces linéaires normés. Espaces à produit scalaire. MATHEMATICS Transformations. bisacsh Inner product spaces fast Normed linear spaces fast Electronic book. Sikorska, Justyna. http://id.loc.gov/authorities/names/no2010059471 Tomás, M. Santos (Maria Santos) https://id.oclc.org/worldcat/entity/E39PBJrwTYjJFkDtJypFQ3CPcP http://id.loc.gov/authorities/names/no2010061277 has work: Norm derivatives and characterizations of inner product spaces (Text) https://id.oclc.org/worldcat/entity/E39PCGBFhDPDcCHwycY4VRJhpP https://id.oclc.org/worldcat/ontology/hasWork Print version: Alsina, Claudi. Norm derivatives and characterizations of inner product spaces. Singapore : World Scientific, ©2010 9789814287265 (OCoLC)540206466 |
| spellingShingle | Alsina, Claudi Norm derivatives and characterizations of inner product spaces / Introduction. Historical notes -- Normed linear spaces -- Strictly convex normed linear spaces -- Inner product spaces --Orthogonalities in normed linear spaces -- Norm derivatives. Norm derivatives : definition and basic properties -- Orthogonality relations based on norm derivatives -- p'[symbol]-orthogonal transformations -- On the equivalence of two norm derivatives -- Norm derivatives and projections in normed linear spaces -- Norm derivatives and Lagrange's identity in normed linear spaces -- On some extensions of the norm derivatives -- p-orthogonal additivity -- Norm derivatives and heights. Definition and basic properties -- Characterizations of inner product spaces involving geometrical properties of a height in a triangle -- Height functions and classical orthogonalities -- A new orthogonality relation -- Orthocenters -- A characterization of inner product spaces involving an isosceles trapezoid property -- Functional equations of the height transform -- Perpendicular bisectors in Normed spaces. Definitions and basic properties -- A new orthogonality relation -- Relations between perpendicular bisectors and classical orthogonalities -- On the radius of the circumscribed circumference of a triangle -- Circumcenters in a triangle -- Euler line in real normed space -- Functional equation of the perpendicular bisector transform -- Bisectrices in real Normed spaces. Bisectrices in real normed spaces -- A new orthogonality relation -- Functional equation of the bisectrix transform -- Generalized bisectrices in strictly convex real normed spaces -- Incenters and generalized bisectrices -- Areas of triangles in Normed spaces. Definition of four areas of triangles -- Classical properties of the areas and characterizations of inner product spaces -- Equalities between different area functions -- The area orthogonality. Normed linear spaces. http://id.loc.gov/authorities/subjects/sh85092434 Inner product spaces. http://id.loc.gov/authorities/subjects/sh85066514 Espaces linéaires normés. Espaces à produit scalaire. MATHEMATICS Transformations. bisacsh Inner product spaces fast Normed linear spaces fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85092434 http://id.loc.gov/authorities/subjects/sh85066514 |
| title | Norm derivatives and characterizations of inner product spaces / |
| title_auth | Norm derivatives and characterizations of inner product spaces / |
| title_exact_search | Norm derivatives and characterizations of inner product spaces / |
| title_full | Norm derivatives and characterizations of inner product spaces / Claudi Alsina, Justyna Sikorska, M Santos Tomás. |
| title_fullStr | Norm derivatives and characterizations of inner product spaces / Claudi Alsina, Justyna Sikorska, M Santos Tomás. |
| title_full_unstemmed | Norm derivatives and characterizations of inner product spaces / Claudi Alsina, Justyna Sikorska, M Santos Tomás. |
| title_short | Norm derivatives and characterizations of inner product spaces / |
| title_sort | norm derivatives and characterizations of inner product spaces |
| topic | Normed linear spaces. http://id.loc.gov/authorities/subjects/sh85092434 Inner product spaces. http://id.loc.gov/authorities/subjects/sh85066514 Espaces linéaires normés. Espaces à produit scalaire. MATHEMATICS Transformations. bisacsh Inner product spaces fast Normed linear spaces fast |
| topic_facet | Normed linear spaces. Inner product spaces. Espaces linéaires normés. Espaces à produit scalaire. MATHEMATICS Transformations. Inner product spaces Normed linear spaces Electronic book. |
| work_keys_str_mv | AT alsinaclaudi normderivativesandcharacterizationsofinnerproductspaces AT sikorskajustyna normderivativesandcharacterizationsofinnerproductspaces AT tomasmsantos normderivativesandcharacterizationsofinnerproductspaces |