Banach–Mazur theorem

In functional analysis, a field of mathematics, the Banach–Mazur theorem is a theorem roughly stating that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named after Stefan Banach and Stanisław Mazur.

Statement

Every real, separable Banach space $(X, ||\cdot||)$ is isometrically isomorphic to a closed subspace of $C 0 ([0, 1], R)$ , the space of all continuous functions from the unit interval into the real line.

Comments

On the one hand, the Banach–Mazur theorem seems to tell us that the seemingly vast collection of all separable Banach spaces is not that vast or difficult to work with, since a separable Banach space is "only" a collection of continuous paths. On the other hand, the theorem tells us that $C 0 ([0, 1], R)$ is a "really big" space, big enough to contain every possible separable Banach space.

Non-separable Banach spaces cannot embed isometrically in the separable space $C 0 ([0, 1], R)$ , but for every Banach space $X$ , one can find a compact Hausdorff space $K$ and an isometric linear embedding $j$ of $X$ into the space $C(K)$ of scalar continuous functions on $K$ . The simplest choice is to let $K$ be the unit ball of the continuous dual $X'$ , equipped with the w*-topology. This unit ball $K$ is then compact by the Banach–Alaoglu theorem. The embedding $j$ is introduced by saying that for every $x \in X$ , the continuous function $j (x)$ on $K$ is defined by

\forall x'\in K:\qquad j(x)(x')=x'(x).

The mapping $j$ is linear, and it is isometric by the Hahn–Banach theorem.

Another generalization was given by Kleiber and Pervin (1969): a metric space of density equal to an infinite cardinal $α$ is isometric to a subspace of $C 0 ([0,1] α, R)$ , the space of real continuous functions on the product of $α$ copies of the unit interval.

Stronger versions of the theorem

Let us write $C k [0, 1]$ for $C k ([0, 1], R)$ . In 1995, Luis Rodríguez-Piazza proved that the isometry $i : X \to C 0 [0, 1]$ can be chosen so that every non-zero function in the image $i (X)$ is nowhere differentiable. Put another way, if $D \subset C 0 [0, 1]$ consists of functions that are differentiable at at least one point of $[0, 1]$ , then $i$ can be chosen so that $i (X) \cap D = {0}.$ This conclusion applies to the space $C 0 [0, 1]$ itself, hence there exists a linear map $i : C 0 [0, 1] \to C 0 [0, 1]$ that is an isometry onto its image, such that image under $i$ of $C 0 [0, 1]$ (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects $D$ only at $0$ : thus the space of smooth functions (with respect to the uniform distance) is isometrically isomorphic to a space of nowhere-differentiable functions. Note that the (metrically incomplete) space of smooth functions is dense in $C 0 [0, 1]$ .

References

Bessaga, Czesław & Pełczyński, Aleksander (1975). Selected topics in infinite-dimensional topology. Warszawa: PWN.
Kleiber, Martin; Pervin, William J. (1969). "A generalized Banach-Mazur theorem". Bull. Austral. Math. Soc. 1 (2): 169–173. doi:10.1017/S0004972700041411 – via Cambridge University Press.
Rodríguez-Piazza, Luis (1995). "Every separable Banach space is isometric to a space of continuous nowhere differentiable functions". Proc. Amer. Math. Soc. 123 (12). American Mathematical Society: 3649–3654. doi:10.2307/2161889. JSTOR 2161889.

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Functional analysis (topics – glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic / Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

Read other articles:

Article 27 de la Constitution de la Cinquième République française

Cet article est une ébauche concernant le droit français. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. Article 27 de la Constitution du 4 octobre 1958 Données clés Présentation Pays France Langue(s) officielle(s) Français Type Article de la Constitution Adoption et entrée en vigueur Législature IIIe législature de la Quatrième République française Gouvernement Charles de Gaulle (3e) Promulgation 4...

Magma (computer algebra system)

MagmaDeveloper(s)Computational Algebra Group, School of Mathematics and Statistics, University of SydneyStable release2.27-8[1] / 22 February 2023; 13 months ago (2023-02-22) Written inC (kernel), Magma (libraries)[2]Operating systemCross-platformTypeComputer algebra systemLicenseCost recovery (non-commercial proprietary)Websitemagma.maths.usyd.edu.au Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combina...

Assassination of John F. Kennedy

1963 murder of the 35th U.S. President Kennedy assassination redirects here. For the assassination of John's brother Robert, see Assassination of Robert F. Kennedy. November 22, 1963 redirects here. For the date, see November 22, 1963 (Friday). Assassination of John F. KennedyPresident John F. Kennedy, his wife Jacqueline, Texas governor John Connally, and Connally's wife Nellie in the presidential limousine minutes before the assassination in DallasLocationDealey Plaza in Dallas, Texas, U.S....

KLUC-FM

Radio station in Las Vegas, NevadaKLUC-FMLas Vegas, NevadaBroadcast areaLas Vegas, NevadaFrequency98.5 MHz (HD Radio)Branding98.5 KLUCProgrammingLanguage(s)EnglishFormatTop 40/CHRSubchannelsHD2: Sports gambling The BetHD3: Regional Mexican Fiesta 98.1OwnershipOwnerAudacy, Inc.(Audacy License, LLC, as Debtor-in-Possession)Sister stationsKMXBKXNTKXQQ-FMHistoryFirst air date1963 (1963)Former call signsKRBO-FM (CP, 1961–62)Call sign meaningLuckyTechnical information[1]Licensing aut...

Fujian–Taiwan relationship

Relations between Taiwan and the mainland Chinese province of Fujian This article is about the relationship between Fujian, a province of the People's Republic of China (PRC) and Taiwan. For the relationship between the PRC itself and the ROC, see Cross-Strait relations. The northwest side of the Taiwan Strait is Fujian, and the southeast side is Taiwan. The Fujian–Taiwan relations, also known as the Min–Tai relations,[1] refers to the relationship between Fujian, which is located...

Food contaminant

Harmful substance in food This article is about the chemical contamination of food. For microbiological contamination, see Foodborne illness. Food safety Terms Foodborne illness Good manufacturing practice (GMP) Hazard analysis and critical control points (HACCP) Hazard analysis and risk-based preventive controls (HARPC) Critical control point Critical factors FAT TOM pH Water activity (aw) Bacterial pathogens Clostridium botulinum Escherichia coli Listeria Salmonella Vibrio cholerae Cronobac...

Boeing 757

Vous lisez un « article de qualité » labellisé en 2018. Boeing 757 Un Boeing 757-200 de Delta Air Lines en approche finale. Rôle Avion de ligne à réaction à fuselage étroit Constructeur Boeing Commercial Airplanes Premier vol 19 février 1982 Mise en service 1er janvier 1983 Premier client Eastern Airlines Client principal Delta Air Lines, United Airlines, American Airlines, UPS Airlines Coût unitaire 757-200 : 65 millions US$ (2002)757-300 : 80 millio...

内华达州

内華達州美國联邦州State of Nevada 州旗州徽綽號：產銀之州、起戰之州地图中高亮部分为内華達州坐标：35°N－42°N, 114°W－120°W国家美國建州前內華達领地加入聯邦1864年10月31日（第36个加入联邦）首府卡森城最大城市拉斯维加斯政府 • 州长（英语：List of Governors of {{{Name}}}]]） • 副州长（英语：List of lieutenant governors of {{{Name}}}]]）喬·隆巴爾多（R斯塔...

Vauxhall 10-4

Motor vehicle Vauxhall Ten1938 Vauxhall Ten 4-Door SaloonOverviewManufacturerVauxhall (General Motors)Also calledVauxhall H-type or HIY-typeVauxhall Wyvern (Australia)[1]Production1937–19401946–1947AssemblyLuton, EnglandTanjung Priok, Batavia, NEI (NV GM Java)Body and chassisClassSmall family car (C)Body style4-door saloon2-door coupé[2]2-door tourer (Australia)[1]2-door roadster (Australia)[1]LayoutFront-engine, rear-wheel-driveRelatedVauxhall ...

Pengepungan Esztergom (1543)

Pengepungan EsztergomTanggal25 Juli – 10 Agustus 1543LokasiEsztergom, HungariaHasil Utsmaniyah merebut EsztergomPihak terlibat Kekaisaran Romawi Suci Kesultanan Utsmaniyah PrancisTokoh dan pemimpin Suleiman IKekuatan Unit Artileri Pengepungan Esztergom(1543) di Hungaria. Pengepungan Esztergom terjadi antara 25 Juli hingga 10 Agustus 1543, ketika Sultan Utsmaniyah, yaitu Suleiman I mengepungepung kota Esztergom modern di Hungaria. Kota itu direbut oleh Turki Utsmani setelah ...

منتخب كوريا الجنوبية لهوكي الجليد للناشئين

منتخب كوريا الجنوبية لهوكي الجليد للناشئين البلد كوريا الجنوبية رمز IIHF KOR مشاركة دولية كوريا الشمالية 10–1 كوريا الجنوبية (آيندهوفن، هولندا؛ 16 مارس 1990) أكبر فوز كوريا الجنوبية 26–1 اليونان (بلغراد، يوغوسلافيا؛ 2 يناير 1991) أكبر هزيمة كوريا الجنوبية 1–13 بريط...

The Logic of Sense

1969 book by Gilles Deleuze The Logic of Sense Cover of the French editionAuthorGilles DeleuzeOriginal titleLogique du sensTranslatorsMark Lester, Charles StivaleLanguageFrenchSeriesEuropean PerspectivesSubjectMeaningPublished 1969 (Les Éditions de Minuit, in French) 1990 (Columbia University Press, in English) Publication placeFranceMedia typePrint (hardcover and paperback)Pages392 (French edition)393 (Columbia University Press edition)ISBN978-0231059831 The Logic of Sense (French...

معاهدة ريو دي جانيرو (1825)

معاهدة ريو دي جانيرومعلومات عامةالنوع معاهدة الموقعون إمبراطورية البرازيل — مملكة البرتغال اللغة البرتغالية تعديل - تعديل مصدري - تعديل ويكي بيانات إمبراطورية البرازيل في 1889. معاهدة ريو دي جانيرو هي معاهدة بين مملكة البرتغال و‌الإمبراطورية البرازيلية، وقعت في 29 أغسطس 182...

Offshore Leaks

Disclosure of tax haven client data in 2013 Note that Offshore Leaks is not part of the WikiLeaks Cablegate leak nor is it part of the Panama Papers. You can help expand this article with text translated from the corresponding article in German. (April 2013) Click [show] for important translation instructions. View a machine-translated version of the German article. Machine translation, like DeepL or Google Translate, is a useful starting point for translations, but translators must revi...

Explorer's Grand Slam

Adventurering goal of Earth's Poles and Seven Summits Sir David Hempleman-Adams standing in Antarctica in 2018. In 1998 he became the first person to complete an Explorer's Grand Slam. Vanessa O'Brien skiing the last degree to the North Pole as part of the Explorer's Grand Slam in 2013. She was the first woman to complete the Last Degree in under one year. The Explorer's Grand Slam is an adventurer goal to reach the North Pole and South Pole, as well as climb the Seven Summits (Everest, Aconc...

Skyscraper

Tall habitable building For other uses, see Skyscraper (disambiguation). Not to be confused with Highrise. Completed in 2009, the Burj Khalifa, in Dubai, United Arab Emirates is currently[update] the tallest building in the world, with a height of 829.8 meters (2,722 ft). The setbacks at various heights are a typical skyscraper feature. A skyscraper is a tall continuously habitable building having multiple floors. Modern sources define skyscrapers as being at least 100 meters (33...

Serie B 1985-1986 (pallacanestro maschile)

Serie B maschile FIP 1985-1986Dettagli della competizioneSport Pallacanestro OrganizzatoreLNP Federazione FIP Periodoottobre 1985 —giugno 1986 Squadre32 (in 2 gironi) VerdettiPromozioni Corona Cremona Scaligera Verona Libertas Pescara Retrocessioni Calabrese Monopoli Pall. Chieti Robur Osimo Pall. Roseto Sutor Montegranaro Basket Scauri N.B. Campobasso Monte Di Procida Cronologia della competizioneed. successiva → &...

Prevlaka

Border Dispute in Europe Not to be confused with Privlaka. For the village near Rugvica, Croatia, see Prevlaka, Zagreb County. Cape Oštro on the Prevlaka peninsula. Prevlaka (Croatian pronunciation: [prɛ̂ʋlaka]) is a small peninsula in southern Croatia, near the border with Montenegro, at the entrance to the Bay of Kotor on the eastern Adriatic coast. Because of its strategic location in the southern Adriatic, in the aftermath of the SFR Yugoslav breakup, the peninsula became subj...

Tanks in the Japanese Army

IJA 4th Armored Division with Type 3 Chi-Nu tanks and Type 3 Ho-Ni III tank destroyers vteHistory of the tankEra World War I Interwar World War II Cold War Post–Cold War Country Australia United Kingdom Cuba China Canada New Zealand Czechoslovakia France Germany Iran Iraq Italy Israel Japan Poland North Korea South Korea Soviet Union Spain Sweden United States Ukraine Type Light tank Medium tank Heavy tank Super-heavy tank Cruiser tank Flame tank Infantry tank Main battle tank Tank destroy...

World War II Allied names for Japanese aircraft

Identification chart of Japanese aircraft Mitsubishi G3M aircraft of the Imperial Japanese Navy were nicknamed Nell by Allied forces during World War II. The World War II Allied names for Japanese aircraft were reporting names, often described as codenames, given by Allied personnel to Imperial Japanese aircraft during the Pacific campaign of World War II. The names were used by Allied personnel to identify aircraft operated by the Japanese for reporting and descriptive purposes. Generally, ...

Atrofi

/profillengkap.com/index.php/article/John Francis Mercer

/profillengkap.com/index.php/article/Free Republic of the Congo

Gurmukhi (blok Unicode)

Wikidata

Neutral Buoyancy Laboratory

1985–86 New York Islanders season

Bishop of Birmingham