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Dissertations / Theses on the topic 'Metric'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Ribeiro, Tiago CaÃla. "Metric homology." Universidade Federal do CearÃ, 2007. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=604.

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Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico<br>CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior<br>No presente trabalho desenvolvemos e aplicamos a teoria de homologia mÃtrica, criada por Jean Paul Brasselet e Lev Birbrair. A cada conjunto semialgÃbrico X associamos uma coleÃÃo de espaÃos vetoriais reais (ou grupos abelianos) {MH_k^ν(X)} _{k є Z} de forma que se Ã dado um outro semialgÃbrico X' que Ã semialgebricamente bi-Lipschitz equivalente a X, entÃo MH_k^ν(X) Ã isomorfo a MH_k^ν(X') para todo k. Assim, a coleÃÃo {MH_k^ν(X)} carrega al

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2

Sidiropoulos, Anastasios. "Computational metric embeddings." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/44712.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.<br>Includes bibliographical references (p. 141-145).<br>We study the problem of computing a low-distortion embedding between two metric spaces. More precisely given an input metric space M we are interested in computing in polynomial time an embedding into a host space M' with minimum multiplicative distortion. This problem arises naturally in many applications, including geometric optimization, visualization, multi-dimensional scaling, network spanners, and the computation of ph

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3

Razafindrakoto, Ando Desire. "Hyperconvex metric spaces." Thesis, Stellenbosch : University of Stellenbosch, 2010. http://hdl.handle.net/10019.1/4106.

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Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.<br>ENGLISH ABSTRACT: One of the early results that we encounter in Analysis is that every metric space admits a completion, that is a complete metric space in which it can be densely embedded. We present in this work a new construction which appears to be more general and yet has nice properties. These spaces subsequently called hyperconvex spaces allow one to extend nonexpansive mappings, that is mappings that do not increase distances, disregarding the properties of the spaces in which they are defined. In particular, theorem

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4

Lazaj, Klotilda. "Metric Preserving Functions." Connect to resource online, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1256915437.

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5

Hussain, Azham. "Metric based evaluation of mobile devices : mobile goal question metric (mGQM)." Thesis, University of Salford, 2012. http://usir.salford.ac.uk/26720/.

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The constant developments in technology of mobile devices are rapidly increasing the demand for applications on these devices. In October 2010, there were 300,000 applications for iPhone available on iTunes store to be downloaded, and as of January 2011, the application store had over 9.9 billion downloads. Moreover, it is reported that the Blackberry phone has been used by more than 28 million people in 91 countries. Demand for Android phones is also increasing and exceeds analysts' expectations. As a result of the rapid evolution of mobile device development, a new challenge has emerged for

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6

Anfinsen, Jarle. "Making substitution matrices metric." Thesis, Norwegian University of Science and Technology, Department of Computer and Information Science, 2005. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9237.

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<p>With the emergence and growth of large databases of information, efficient methods for storage and processing are becoming increasingly important. The existence of a metric distance measure between data entities enables efficient index structures to be applied when storing the data. Unfortunately, this is often not the case. Amino acid substitution matrices, which are used to estimate similarities between proteins, do not yield metric distance measures. Finding efficient methods for converting a non-metric matrix into a metric one is therefore highly desirable. In this work, the problem of

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7

Bagge, Joar. "A graphotactic language metric." Thesis, KTH, Matematik (Inst.), 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-128781.

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In this bachelor’s thesis, we try to classify and identify written human languages by studying the ordering of letters in text. Automatic language identification is of interest in areas such as text indexing, machine translation and natural language parsing. Eleven written languages which use the Latin alphabet are considered and modelled with a Markov chain on the letter level. Texts from the New Testament and Wikipedia are used as training data. The distances between the languages are then measured by using a matrix-based metric on the transition matrices, and visualized in a dendrogram. A p

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8

Matthews, S. G. "Metric domains for completeness." Thesis, University of Warwick, 1985. http://wrap.warwick.ac.uk/60775/.

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Completeness is a semantic non-operational notion of program correctness suggested (but not pursued) by W.W.Wadge. Program verification can be simplified using completeness, firstly by removing the approximation relation from proofs, and secondly by removing partial objects from proofs. The dissertation proves the validity of this approach by demonstrating how it can work in the class of metric domains. We show how the use of Tarski's least fixed point theorem can be replaced by a non-operational unique fixed point theorem for many well behaved Programs. The proof of this theorem is also non-o

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9

Al-Harbi, Sami. "Clustering in metric spaces." Thesis, University of East Anglia, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.396604.

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10

Jensen, Harold Franklin. "Variable buoyancy system metric." Thesis, Massachusetts Institute of Technology, 2009. http://hdl.handle.net/1721.1/58193.

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Thesis (S.M.)--Joint Program in Applied Ocean Science and Engineering (Massachusetts Institute of Technology, Dept. of Mechanical Engineering; and the Woods Hole Oceanographic Institution), 2009.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 111-112).<br>Over the past 20 years, underwater vehicle technology has undergone drastic improvements, and vehicles are quickly gaining popularity as a tool for numerous oceanographic tasks. Systems used on the vehicle to alter buoyancy, or variable buoyancy (VB) systems, have seen only minor improvements during the sa

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11

Nordebo, Jonatan. "The Reissner-Nordström metric." Thesis, Umeå universitet, Institutionen för fysik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-118337.

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A brief review of special and general relativity including some classical electrodynamics is given. We then present a detailed derivation of the Reissner-Nordström metric. The derivation is done by solving the Einstein-Maxwell equations for a spherically symmetric electrically charged body. The physics of this spacetime is then studied. This includes gravitational time dilation and redshift, equations of motion for both massive and massless non-charged particles derived from the geodesic equation and equations of motion for a massive charged particle derived with lagrangian formalism. Finally,

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12

Choy, Tze Leung. "Sparse distance metric learning." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:a98695a3-0a60-448f-9ec0-63da3c37f7fa.

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A good distance metric can improve the accuracy of a nearest neighbour classifier. Xing et al. (2002) proposed distance metric learning to find a linear transformation of the data so that observations of different classes are better separated. For high-dimensional problems where many un-informative variables are present, it is attractive to select a sparse distance metric, both to increase predictive accuracy but also to aid interpretation of the result. In this thesis, we investigate three different types of sparsity assumption for distance metric learning and show that sparse recovery is pos

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13

Fuhry, David P. "Skylines in Metric Space." Kent State University / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=kent1208562156.

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14

Büch, Lutz [Verfasser], and Artur [Akademischer Betreuer] Andrzejak. "Metric Selection and Metric Learning for Matching Tasks / Lutz Büch ; Betreuer: Artur Andrzejak." Heidelberg : Universitätsbibliothek Heidelberg, 2020. http://d-nb.info/1224684524/34.

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15

Laub, Julian. "Non-metric pairwise proximity data." [S.l.] : [s.n.], 2004. http://edocs.tu-berlin.de/diss/2004/laub_julian.pdf.

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16

Lee, Seunghwan Han. "Probabilistic reasoning on metric spaces." [Bloomington, Ind.] : Indiana University, 2009. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3380096.

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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics and Cognitive Science, 2009.<br>Title from PDF t.p. (viewed on Jul 19, 2010). Source: Dissertation Abstracts International, Volume: 70-12, Section: B, page: 7604. Adviser: Lawrence S. Moss.

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17

Chang, Hong. "Semi-supervised distance metric learning /." View abstract or full-text, 2006. http://library.ust.hk/cgi/db/thesis.pl?COMP%202006%20CHANG.

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18

Rekdal, Espen Ekornes. "Metric Indexing in Time Series." Thesis, Norwegian University of Science and Technology, Department of Computer and Information Science, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-10487.

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19

Erlandsen, Stian. "Metric indexing by database techniques." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for datateknikk og informasjonsvitenskap, 2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-12565.

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Similarity search is very useful in many applications. Because of the complex-ity and expensive nature of such search operations, many existing methods re-quire special access methods and cannot be directly integrated with commercialDBMSs(Database Management Systems). NTNUStore is a framework to aid re-search in this eld and focuses on disk-based metric indexing to keep it compatiblewith commercial DBMSs.This project has implemented and experimented with a version of the LAESA(Linear Approximating and Eliminating Search Algorithm) in NTNUStore. Theresults are close to what was represented in E

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20

Lemaire-Beaucage, Jonathan. "Voronoi Diagrams in Metric Spaces." Thesis, Université d'Ottawa / University of Ottawa, 2012. http://hdl.handle.net/10393/20736.

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In this thesis, we will present examples of Voronoi diagrams that are not tessellations. Moreover, we will find sufficient conditions on subspaces of E2, S2 and the Poincaré disk and the sets of sites that guarantee that the Voronoi diagrams are pre-triangulations. We will also study g-spaces, which are metric spaces with ‘extendable’ geodesics joining any 2 points and give properties for a set of sites in a g-space that again guarantees that the Voronoi diagram is a pre-triangulation.

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21

Li, Rong. "A system-wide anonymity metric." Thesis, Wichita State University, 2011. http://hdl.handle.net/10057/5041.

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In this thesis we present a critical analysis of the system-wide anonymity metric by Edman et al. [1], which is based on the permanent value of a doubly-stochastic matrix. From the view of intuitive understanding, we show that a metric that looks no further than the permanent, a composite value, is at best a rough indicator of anonymity, We find the range where its inaccuracy is acute, and propose a better anonymity indicator. Also, we show that this metric fails to possess desirable generalization properties by constructing an information-preserving embedding of a smaller class of attacks int

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22

Chandirasekaran, Devasena. "Optimized AODV with modified metric." Thesis, Wichita State University, 2011. http://hdl.handle.net/10057/5162.

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The growth of wireless technology has developed since the invention of wireless devices. With the recent introduction of smart phones and other latest mobile devices, the growing need for networking is limitless. This calls for a more efficient Adhoc network formed on the go. The goal is to improve the MANET type of Adhoc networks. This infrastructure less network has routing protocols to route the packets to the intended recipient in an organized manner. AODV protocol chooses the best path to the destination based on the hop count as metric it has no power, stability and bandwidth optimizatio

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23

Khan, Moazzam. "Security metric based risk assessment." Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/47527.

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Modern day computer networks have become very complex and attackers have benefited due to this complexity and have found vulnerabilities and loopholes in the network architecture. In order to identify the attacks from an attacker all aspects of network architecture needs to be carefully examined such as packet headers, network scans, versions of applications, network scans, network anomalies etc. and after the examination attributes playing a significant impact on the security posture of the organization needs to be highlighted so that resources and efforts are directed towards those attribute

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24

Yusupova, Tatiana. "Problems in metric diophantine approximations." Thesis, University of York, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.534907.

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25

Holland, Andrew J. "Metric implementation in U.S. construction." Thesis, Gainesville, Florida : University of Florida, 1997. http://hdl.handle.net/10945/26631.

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Throughout the history of industrialization, people have been trying to develop a single system of measurement with a worldwide acceptance. Since its inception two centuries ago, the metric system has been gaining ground over the traditional "inch pound" system. Today, the United States remains as the sole user of the "inch pound" system of measurements. The pressure for unification into the metric system has been mounting. With the emergence of the "global economy," non metric products are becoming increasingly unacceptable in the world markets. Individual industries, societies and organizati

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26

Rowe, Paul Michael Dominic. "Contributions to metric number theory." Thesis, Royal Holloway, University of London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.408263.

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27

Coffey, Michael R. "Ricci flow and metric geometry." Thesis, University of Warwick, 2015. http://wrap.warwick.ac.uk/67924/.

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This thesis considers two separate problems in the field of Ricci flow on surfaces. Firstly, we examine the situation of the Ricci flow on Alexandrov surfaces, which are a class of metric spaces equipped with a notion of curvature. We extend the existence and uniqueness results of Thomas Richard in the closed case to the setting of non-compact Alexandrov surfaces that are uniformly non-collapsed. We complement these results with an extensive survey that collects together, for the first time, the essential topics in the metric geometry of Alexandrov spaces due to a variety of authors. Secondly,

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28

van, Staden Wernd Jakobus. "Metric aspects of noncommutative geometry." Diss., University of Pretoria, 2019. http://hdl.handle.net/2263/77893.

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We study noncommutative geometry from a metric point of view by constructing examples of spectral triples and explicitly calculating Connes's spectral distance between certain associated pure states. After considering instructive nite-dimensional spectral triples, the noncommutative geometry of the in nite-dimensional Moyal plane is studied. The corresponding spectral triple is based on the Moyal deformation of the algebra of Schwartz functions on the Euclidean plane.<br>Dissertation (MSc)--University of Pretoria, 2019.<br>Physics<br>MSc<br>Unrestricted

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29

Pan, Jiajun. "Metric learning for structured data." Thesis, Nantes, 2019. http://www.theses.fr/2019NANT4076.

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L’apprentissage à distance métrique est une branche de l’apprentissage par re-présentation des algorithmes d’apprentissage automatique. Nous résumons le développement et la situation actuelle de l’algorithme actuel d’apprentissage à distance métrique à partir des aspects de la base de données plate et de la base de données non plate. Pour une série d’algorithmes basés sur la distance de Mahalanobis pour la base de données plate qui ne parvient pas à exploiter l’intersection de trois dimensions ou plus, nous proposons un algorithme d’apprentissage métrique basé sur la fonction sousmodulaire. Po

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30

Erninger, Klas. "Algebraic Simplifications of Metric Information." Thesis, KTH, Matematik (Avd.), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-277744.

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This thesis is about how to interpret metric data with topological tools, such as homology. We show how to go from a metric space to a topological space via Vietoris-Rips complexes. We use the usual approach to Topological Data Analysis (TDA), and transform our metric space into tame parametrised vector spaces. It is then shown how to simplify tame parametrised vector spaces. We also present another approach to TDA, where we transform our metric space into a filtrated tame parametrised chain complex. We then show how to simplify chain complexes over fields in order to simplify tame parametrise

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31

Otafudu, Olivier Olela. "Convexity in quasi-metric spaces." Doctoral thesis, University of Cape Town, 2012. http://hdl.handle.net/11427/10950.

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Includes abstract.<br>Includes bibliographical references.<br>The principal aim of this thesis is to investigate the existence of an injective hull in the categories of T-quasi-metric spaces and of T-ultra-quasi-metric spaces with nonexpansive maps.

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32

Okutan, Osman Berat. "Persistence, Metric Invariants, and Simplification." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1559312147225384.

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33

Reinke, Kristen Nicole. "SAFETY METRIC THE WEITZ COMPANY." Thesis, The University of Arizona, 2009. http://hdl.handle.net/10150/192951.

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34

Abou-Moustafa, Karim. "Metric learning revisited: new approaches for supervised and unsupervised metric learning with analysis and algorithms." Thesis, McGill University, 2012. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=106370.

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In machine learning one is usually given a data set of real high dimensional vectors X, based on which it is desired to select a hypothesis θ from the space of hypotheses Θ using a learning algorithm. An immediate assumption that is usually imposed on X is that it is a subset from the very general embedding space Rp which makes the Euclidean distance ∥•∥2 to become the default metric for the elements of X. Since various learning algorithms assume that the input space is Rp with its endowed metric ∥•∥2 as a (dis)similarity measure, it follows that selecting hypothesis θ becomes intrinsically ti

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35

Powell, J. E. "Metric versus non-metric skeletal traits : which is the more reliable indicator of genetic distance?" Thesis, University of Bristol, 1989. http://hdl.handle.net/1983/e44b5162-0f9d-4f0a-afdf-2eac49563e4b.

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36

Gupta, Kishan. "Metric and non-metric inputs influence spatial and working memory processes of medial entorhinal neurons." Thesis, Boston University, 2013. https://hdl.handle.net/2144/12771.

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Thesis (Ph.D.)--Boston University<br>The medial entorhinal cortex (MEC) contains spatial cell types including grid and head direction (HD) cells. Grid cells fire action potentials when animals pass through environmental locations that form vertices of tessellating triangles. HD cells fire when animals face a preferred direction along their azimuth. These cells have been widely studied for their potential metric role in spatial navigation, but considerably less is known about their non-metric functions. This thesis examines non-metric influences on the MEC during working memory maintenance, 'lo

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37

Lease, Loren Rosemond. "Ancestral determination of African American and European American deciduous dentition using metric and non-metric analysis." Columbus, OH : Ohio State University, 2003. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1054742334.

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Thesis (Ph. D.)--Ohio State University, 2003.<br>Title from first page of PDF file. Document formatted into pages; contains xvii, 421 p.: ill. (some col.). Includes abstract and vita. Advisor: Paul W. Sciulli, Dept. of Anthropology. Includes bibliographical references (p. 130-152).

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38

Calisti, Matteo. "Differential calculus in metric measure spaces." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21781/.

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L'obbiettivo di questa tesi è la definizione del calcolo differenziale e dell'operatore di Laplace in spazi metrici di misura. Nel primo capitolo vengono introdotte le definizioni e proprietà principali degli spazi metrici di misura mentre nel secondo quelle riguardanti le funzioni lipschitziane e la derivata metrica di curve assolutamente continue. Nel terzo capitolo quindi viene definito il concetto di p-supergradiente debole e di conseguenza la classe di Sobolev S^p. Nel quarto capitolo viene poi studiata la generalizzazione del concetto di differenziale di f applicato al gradiente di g che

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39

Neri, Giulio. "Effective metric for bootstrapped Newtonian sources." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21717/.

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In this work, we address the problem of constructing an effective space-time picture for the gravitational field in the bootstrapped Newtonian theory (BNT). After a short review of the singularity problem and the corpuscular model of black holes, we present a detailed introduction to the BNT. This theory provides us with an exact formula for the gravitational potential felt by slowly-moving particles outside a static and spherically symmetric source. Our goal is to determine the most general metric tensor that reproduces this potential in the Newtonian limit and use it in relativistic or stro

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Amato, Giuseppe. "Approximate similarity search in metric spaces." [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=964997347.

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Grossman, Mary Alice. "Validation of a Quality Management Metric." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 2000. http://handle.dtic.mil/100.2/ADA384217.

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Thesis (M.S. in Software Engineering) Naval Postgraduate School, Sept. 2000.<br>"September 2000." Thesis advisor(s): Osmundson, John; Michael, J. Bret. Includes bibliographical references (p. 121). Also available online.

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Fevang, Rune, and Arne Bergene Fossaa. "Empirical evaluation of metric indexing methods." Thesis, Norwegian University of Science and Technology, Department of Computer and Information Science, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-8902.

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<p>Metric indexing is a branch of search technology that is designed for search non-textual data. Examples of this includes image search (where the search query is an image), document search (finding documents that are roughly equal) to search in high-dimensional Euclidean spaces. Metric indexing is based on the theory of metric spaces, where the only thing known about a set of objects is the distance between them (defined by a metric distance function). A large number of methods have been proposed to solve the metric indexing problem. In this thesis, we have concentrated on new approaches to

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43

Palmer, Ian Christian. "Riemannian geometry of compact metric spaces." Diss., Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/34744.

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A construction is given for which the Hausdorﬀ measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorﬀ measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorﬀ and box dimensions diﬀer---in particular, it does n

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44

Nomura, Kouichi. "A recipe for multi-metric gravity." 京都大学 (Kyoto University), 2015. http://hdl.handle.net/2433/199102.

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45

Jones, Miranda Rose. "Conformal deformation of a conic metric." Thesis, Wichita State University, 2011. http://hdl.handle.net/10057/3996.

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The problem studied here focuses on a compact manifold M without boundary in which the Riemannian metric g is on Λ = M – {p1, p2,…,pκ}. Near the pi 's, g has a particular type of singularity in which locally M = (0, δ)x × Ywhere Y is a Riemannian manifold with metric h. Calculation techniques involving Christoffel symbols, scalar curvature, and the Lapalacian of the manifold are used to reduce the Yamabe equation to a system of partial differential equations. After assuming that a function u > 0 satisfying the Yamabe equation exists, the most singular partial differential equation is solved

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46

Jägrell, Linus. "Geometry of the Lunin-Maldacena metric." Thesis, KTH, Teoretisk fysik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-153502.

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47

Witt, Frederik. "Special metric structures and closed forms." Thesis, University of Oxford, 2005. http://ora.ox.ac.uk/objects/uuid:30b7a34b-cc46-4981-aee5-964787c1235e.

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In recent work, N. Hitchin described special geometries in terms of a variational problem for closed generic $p$-forms. In particular, he introduced on 8-manifolds the notion of an integrable $PSU(3)$-structure which is defined by a closed and co-closed 3-form. In this thesis, we first investigate this $PSU(3)$-geometry further. We give necessary conditions for the existence of a topological $PSU(3)$-structure (that is, a reduction of the structure group to $PSU(3)$ acting through its adjoint representation). We derive various obstructions for the existence of a topological reduction to $PSU(3

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48

Velani, Sanju Lalji. "Metric diophantine approximation in hyperbolic space." Thesis, University of York, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.304351.

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49

Salp, Cem. "Metric diophantine approximation and cantor sets." Thesis, University of York, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444750.

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Kilbane, James. "Finite metric subsets of Banach spaces." Thesis, University of Cambridge, 2019. https://www.repository.cam.ac.uk/handle/1810/288272.

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The central idea in this thesis is the introduction of a new isometric invariant of a Banach space. This is Property AI-I. A Banach space has Property AI-I if whenever a finite metric space almost-isometrically embeds into the space, it isometrically embeds. To study this property we introduce two further properties that can be thought of as finite metric variants of Dvoretzky's Theorem and Krivine's Theorem. We say that a Banach space satisfies the Finite Isometric Dvoretzky Property (FIDP) if it contains every finite subset of $\ell_2$ isometrically. We say that a Banach space has the Finite

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