Retrospective study of endodontic treatment and retreatment

Orientador: Francisco José de Souza FilhoTese (doutorado) - Universidade Estadual de Campinas. Faculdade de Odontologia de PiracicabaResumo: A proposta deste estudo foi avaliar o índice de sucesso e insucesso do tratamento endodôntico original realizado por um especialista. Um total de 1376 dentes foi examinado clínica e radiograficamente e os resultados analisados estatisticamente pelos testes exatos de Pearson ou Fisher e regressão logística multivariável. Todos os testes estatísticos foram realizados com 5% de significância. Dos 1376 dentes que receberam tratamentos endodônticos originais, o índice de sucesso foi de 94%. .A análise multivarável identificou a presença de complicações trans-operatórias (fratura de lima, perfuração e flare-up), assim como a ausência de restauração definitiva durante as proservações como fatores que influíram negativamente nos resultados dos tratamentos endodônticos originais. Os 6% (83 dentes) considerados fracassos receberam específicas intervenções ou simplesmente foram proservados por um tempo maior (na ausência de sinais e sintomas). Decorridos 5-33 anos, a segunda proservação foi feita, e os resultados mostraram índice de sucesso de 72.7% dos dentes submetidos ao retratamento endodôntico. Em caso de fracasso do tratamento endodôntico original, a reintervenção endodôntica apresentou superioridade estatística em relação à complementação com procedimentos cirúrgicos (parendodôntica ou periodontal)Abstract: The purpose of this study was to evaluate the original endodontic treatment outcome performed by an endodontic specialist. A total of 1,376 teeth were examined clinically and radiographically and the results were analyzed statistically by Pearson or Fisher's test and multivariate logistic regression. All tests were performed as two tailed and interpreted at a 5% significance level. The overall endodontic rate was 94%. Multivariate analysis identified the presence of procedural complications (file breakage, perforation and flare-up), as well as the absence of the restorations at follow-ups as the significant predictors of outcome, showing lower rates of success. The 6% (83 teeth) considered failures either received a specific intervention or were simply monitored for a long period (if a reduction of the periapical lesion was evident or signs and symptoms were absent). After at least 5 years or longer, these teeth were again followed-up and the results showed success rate of 72.7%. In cases of original endodontic failure, the nonsurgical endodontic retreatment showed a statistically higher success rate than retreated teeth complemented with periapical surgery or periodontal treatmentDoutoradoEndodontiaDoutor em Clínica Odontológic

Fatores de sucesso em endodontia : analise retrospectiva de 2.000 casos clinicos

Orientador: Francisco Jose de Souza FilhoDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Odontologia de PiracicabaResumo: Este trabalho retrospectivo avaliou a porcentagem de sucesso / insucesso de 15.617 tratamentos e retratamentos endodônticos realizados em 8.590 pacientes atendidos no consultório particular entre março de 1971 a março de 2000. O mínimo de 18 meses entre o tratamento e o controle era necessário para que o caso pudesse ser incluído nesta pesquisa. Os controles clínico-radiográficos foram realizados nos pacientes que nos prazos pré-determinados retomavam ao consultório para fazer a proservação ou quando quando havia necessidade de submeter-se a uma nova terapia endodôntica, ocasião em que os controles dos dentes já tratados eram feitos. No exame clínico, realizado pelo operador, eram observadas as presenças ou ausências de sinais ou sintomas, e obtida a radiografia de controle. As radiografias tiradas durante o tratamento e de controle foram analisadas por dois observadores independentes, sendo que o primeiro analisou as radiografias pelo menos duas vezes com um intervalo de 3 meses e o segundo observador analisou as radiografias no final do trabalho. Na ausência de concordância entre os observadores, os casos foram discutidos em conjunto até se chegar a um consenso. O tratamento endodôntico foi considerado sucesso quando as seguintes condições eram encontradas: ausência de sinais e sintomas clínicos e radiograficamente, ausência de lesão óssea ou reparo de uma pré-existente. Os dados obtidos foram transformados em porcentagem e examinados estatisticamente usando o teste qui-quadrado de Pearson e teste exato de Fisher. Entre os vários fatores que pudessem interferir no índice de sucesso / insucesso, 5 fatores foram analisados: estados pulpar e peri-radicular, modalidade de tratamentos, número de sessões operatórias e nível apical de obturação. A média geral de sucesso foi de 91,45%, considerando os tratamentos e retratamentos endodônticos realizados. Dos fatores analisados, a condição pulpar (polpa com vitalidade), da região peri-radicular (ausência de lesão), modalidade de intervenção (convencional), número de sessões (sessão única) e nível apical de obturação (aquém ápice) apresentaram índices de sucesso estatisticamente maiores do que polpas sem vitalidade, com lesão, retratamento, sessões múltiplas e obturações no limite radiográfico ou além ápiceAbstract: The c1inical and radiographic follow-up examinations were performed to evaluate the percentage of success and failure of endodontic therapy in 2000 teeth. For inc1usion in this retrospective survey, a review at 18 months or longer was necessary after endodontic or nonsurgical treatment had been performed. The follow-ups sessions were carried out on patients that returned to the dental office in pre-scheduled periods or needed to go through other treatments. For teeth examined more than once, only the findings of the final examination were considered. From 8,590 patients and 15,617 endodontic therapies performed, this survey was considered complete when it reached the total of 2,000 followed-up cases. AlI the recorded information !Tom the files was transfered to a computerized database. The clinical and radiographic folow-up examinations were performed by the primary author. Clinically, no loss of function and absence of signs and symptoms were considered success. The radiographic examination was analyzed separately by two independent observers; the primary author evaluated twice with an interval of at least 3 months between evaluations; the second observer evaluated once at the end of the survey. In cases of desagreement, the observers analyzed them together to establish whether they were success or failure. Radiographically, the case was considered success when the periodontalligament space was normal on the original diagnostic radiograph andt remained unchanged on the recall radiograph, or there was healing of a radiolucent area visible on the original preoperative radiograph and the periodontalligament space had returned to normal. Teeth with open apex, injuried with luxation, intrusion, extrusion, avulsion, horizontal or vertical !Tactures and teeth requiring endodontic surgery were exc1uded !Tom this study. Among many factors that could interfere in the outcome of endodontic therapy, 5 variables were analyzed: the pulp and periapical status, the treatment modality (conventional or retreatment), number of appointments and the leveI of filling. The treatment and retreatment results regarding specific preoperative, intraoperative and postopeartive data were analyzed statistically using Chi-square or Fisher tests with a 5% leveI of significance to determine whether any factors affected the success or failure rates. The overall success rate, considering both conventional and nonsurgical treatment was 91,45%. The results showed that vital teeth, teeth without lesion, conventional treatment, single appointment and short filling leveI had a statistically higher success rate than nonvital teeth, teeth with lesion, nonsurgical treatment, multiple appointment and filling material to the apex level or with excessMestradoEndodontiaMestre em Clínica Odontológic

早期糖尿病性腎症での腎臓におけるα-Klotho 発現の低下とその尿中カルシウム排世に対する役割についての検討

Hypercalciuria is one of the early manifestations of diabetic nephropathy. We explored here the role of α-Klotho, a protein expressed predominantly in distal convoluted tubules that has a role in calcium reabsorption. We studied 31 patients with early diabetic nephropathy and compared them with 31 patients with IgA nephropathy and 7 with minimal change disease. Renal α-Klotho expression was significantly lower and urinary calcium excretion (UCa/UCr) significantly higher in diabetic nephropathy than in IgA nephropathy or minimal change disease. Multiple regression analyses indicated that α-Klotho mRNA was inversely correlated with calcium excretion. We next measured these parameters in a mouse model of streptozotocin (STZ)-induced diabetic nephropathy, characterized by glomerular hyperfiltration, as seen in early diabetic nephropathy. We also confirmed a reduction of renal α-Klotho mRNA down to almost 50% and enhanced calcium excretion in mice with STZ-induced diabetic nephropathy in comparison with nondiabetic mice. Hypercalciuria was exacerbated in heterozygous α-Klotho knockout mice in comparison with wild-type mice, each with STZ-induced diabetic nephropathy. Thus, α-Klotho expression was decreased in distal convoluted tubules in diabetic nephropathy in humans and mice. Renal loss of α-Klotho may affect urinary calcium excretion in early diabetic nephropathy.博士（医学）・乙第1293号・平成24年5月28日© 2012 International Society of Nephrolog

Partial Differentiation, Differentiation and Continuity on n-Dimensional Real Normed Linear Spaces

In this article, we aim to prove the characterization of differentiation by means of partial differentiation for vector-valued functions on n-dimensional real normed linear spaces (refer to [15] and [16]).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanNaumowicz Adam - Institute of Computer Science, University of Białystok, Akademicka 2, 15-267 Białystok, PolandEndou Noboru - Nagano National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Laurent Schwartz. Cours d'analyse. Hermann, 1981. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

The Differentiable Functions from R into Rⁿ

In control engineering, differentiable partial functions from R into Rⁿ play a very important role. In this article, we formalized basic properties of such functions.Narita Keiko - Hirosaki-city, Aomori, JapanKorniłowicz Artur - Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Takao Inoué, Adam Naumowicz, Noboru Endou, and Yasunari Shidama. Partial differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 19(1):1-9, 2011, doi: 10.2478/v10037-011-0001-x.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Keiko Narita, Artur Kornilowicz, and Yasunari Shidama. More on the continuity of real functions. Formalized Mathematics, 19(4):233-239, 2011, doi: 10.2478/v10037-011-0032-3.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Hiroyuki Okazaki, Noboru Endou, Keiko Narita, and Yasunari Shidama. Differentiable functions into real normed spaces. Formalized Mathematics, 19(2):69-72, e2011, doi: 10.2478/v10037-011-0012-7.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011, doi: 10.2478/v10037-011-0008-3.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanEndou Noboru - Gifu National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Yatsuka Nakamura, Artur Korniłowicz, Nagato Oya, and Yasunari Shidama. The real vector spaces of finite sequences are finite dimensional. Formalized Mathematics, 17(1):1-9, 2009, doi:10.2478/v10037-009-0001-2.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Laurent Schwartz. Cours d'analyse. Hermann, 1981. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Differentiable Functions into Real Normed Spaces

In this article, we formalize the differentiability of functions from the set of real numbers into a normed vector space [14].Okazaki Hiroyuki - Shinshu University, Nagano, JapanEndou Noboru - Nagano National College of Technology, Nagano, JapanNarita Keiko - Hirosaki-city, Aomori, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55- 65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011, doi: 10.2478/v10037-011-0008-3.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Laurent Schwartz. Cours d'analyse, vol. 1. Hermann Paris, 1967. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

Partial Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanNaumowicz Adam - Institute of Computer Science, University of Białystok, Akademicka 2, 15-267 Białystok, PolandEndou Noboru - Gifu National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces n. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Takao Inoué, Noboru Endou, and Yasunari Shidama. Differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 18(4):207-212, 2010, doi: 10.2478/v10037-010-0025-7.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.Yatsuka Nakamura, Artur Korniłowicz, Nagato Oya, and Yasunari Shidama. The real vector spaces of finite sequences are finite dimensional. Formalized Mathematics, 17(1):1-9, 2009, doi:10.2478/v10037-009-0001-2.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Laurent Schwartz. Cours d'analyse. Hermann, 1981. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki, Yoshinori Fujisawa, and Yatsuka Nakamura. On replace function and swap function for finite sequences. Formalized Mathematics, 9(3):471-474, 2001

Differentiable Functions on Normed Linear Spaces

In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(x + t · h)|| is bounded for t between 0 and 1 by some constant M, then ||ƒ(x + t · h) - ƒ(x)|| ≤ M · ||h||. This theorem is called the mean value theorem for vector-valued functions. By this theorem, the relation between the (total) derivative and the partial derivatives of a function is derived [23].Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. The product space of real normed spaces and its properties. Formalized Mathematics, 15(3):81-85, 2007, doi:10.2478/v10037-007-0010-y.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics, 3(2):235-240, 1992.Hiroyuki Okazaki, Noboru Endou, Keiko Narita, and Yasunari Shidama. Differentiable functions into real normed spaces. Formalized Mathematics, 19(2):69-72, 2011, doi: 10.2478/v10037-011-0012-7.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Laurent Schwartz. Cours d'analyse, vol. 1. Hermann Paris, 1967.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992