Banach Algebra of Bounded Complex-Valued Functionals

In this article, we describe some basic properties of the Banach algebra which is constructed from all bounded complex-valued functionals.Kanazashi Katuhiko - Shizuoka High School, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanJózef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 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.Noboru Endou. Banach algebra of bounded complex linear operators. Formalized Mathematics, 12(3):237-242, 2004.Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93-102, 2004.Noboru Endou. Complex valued functions space. Formalized Mathematics, 12(3):231-235, 2004.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex functions. Formalized Mathematics, 9(1):179-184, 2001.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Yasunari Shidama, Hikofumi Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008, doi:10.2478/v10037-008-0017-z.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.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

The Ck Space

In this article, we formalize continuous differentiability of realvalued functions on n-dimensional real normed linear spaces. Next, we give a definition of the Ck space according to [23].This work was supported by JSPS KAKENHI 22300285Kanazashi Katuhiko - Shizuoka City, JapanOkazaki Hiroyuki - Shinshu University Nagano, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 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 Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Higher-order partial differentiation. Formalized Mathematics, 20(2):113-124, 2012. doi:10.2478/v10037-012-0015-z.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1): 35-40, 1990.Takao Inou´e, 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.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.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.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 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.Kosaku Yosida. Functional Analysis. Springer Classics in Mathematics, 1996

Mitochondrial DNA Variants at Low-Level Heteroplasmy and Decreased Copy Numbers in Chronic Kidney Disease (CKD) Tissues with Kidney Cancer

The human mitochondrial genome (mtDNA) is a circular DNA molecule with a length of 16.6 kb, which contains a total of 37 genes. Somatic mtDNA mutations accumulate with age and environmental exposure, and some types of mtDNA variants may play a role in carcinogenesis. Recent studies observed mtDNA variants not only in kidney tumors but also in adjacent kidney tissues, and mtDNA dysfunction results in kidney injury, including chronic kidney disease (CKD). To investigate whether a relationship exists between heteroplasmic mtDNA variants and kidney function, we performed ultra-deep sequencing (30,000×) based on long-range PCR of DNA from 77 non-tumor kidney tissues of kidney cancer patients with CKD (stages G1 to G5). In total, this analysis detected 697 single-nucleotide variants (SNVs) and 504 indels as heteroplasmic (0.5% ≤ variant allele frequency (VAF) \u3c 95%), and the total number of detected SNVs/indels did not differ between CKD stages. However, the number of deleterious low-level heteroplasmic variants (pathogenic missense, nonsense, frameshift and tRNA) significantly increased with CKD progression

Variant Spectrum of von Hippel-Lindau Disease and Its Genomic Heterogeneity in Japan

Von Hippel-Lindau (VHL) disease is an autosomal dominant, inherited syndrome with variants in the VHL gene, causing predisposition to multi-organ neoplasms with vessel abnormality. Germline variants in VHL can be detected in 80-90% of patients clinically diagnosed with VHL disease. Here, we summarize the results of genetic tests for 206 Japanese VHL families, and elucidate the molecular mechanisms of VHL disease, especially in variant-negative unsolved cases. Of the 206 families, genetic diagnosis was positive in 175 families (85%), including 134 families (65%) diagnosed by exon sequencing (15 novel variants) and 41 (20%) diagnosed by multiplex ligation-dependent probe amplification (MLPA) (one novel variant). The deleterious variants were significantly enriched in VHL disease Type 1. Interestingly, five synonymous or non-synonymous variants within exon 2 caused exon 2 skipping, which is the first report of exon 2 skipping caused by several missense variants. Whole genome and target deep sequencing analysis were performed for 22 unsolved cases with no variant identified and found three cases with VHL mosaicism (variant allele frequency: 2.5-22%), one with mobile element insertion in the VHL promoter region, and two with a pathogenic variant of BAP1 or SDHB. The variants associated with VHL disease are heterogeneous, and for more accuracy of the genetic diagnosis of VHL disease, comprehensive genome and DNA/RNA analyses are required to detect VHL mosaicism, complicated structure variants and other related gene variants

Functional Space C(ω), C0(ω)

In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.Kanazashi Katuhiko - Shizuoka City, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 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. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.Noboru Endou. Banach algebra of bounded complex linear operators. Formalized Mathematics, 12(3):237-242, 2004.Noboru Endou. Banach space of absolute summable complex sequences. Formalized Mathematics, 12(2):191-194, 2004.Noboru Endou. Complex Banach space of bounded linear operators. Formalized Mathematics, 12(2):201-209, 2004.Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93-102, 2004.Noboru Endou. Complex linear space of complex sequences. Formalized Mathematics, 12(2):109-117, 2004.Noboru Endou. Complex valued functions space. Formalized Mathematics, 12(3):231-235, 2004.Noboru Endou. Continuous functions on real and complex normed linear spaces. Formalized Mathematics, 12(3):403-419, 2004.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Katuhiko Kanazashi, Hiroyuki Okazaki, and Yasunari Shidama. Banach algebra of bounded complex-valued functionals. Formalized Mathematics, 19(2):121-126, 2011, doi: 10.2478/v10037-011-0019-0.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Chanapat Pacharapokin, Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Complex function differentiability. Formalized Mathematics, 17(2):67-72, 2009, doi: 10.2478/v10037-009-0007-9.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Yasunari Shidama, Hikofumi Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008, doi:10.2478/v10037-008-0017-z.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.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