Database degli esercizi — book index

English

• \(\overline A\) , see closure .
• \(\partial A\) , see boundary .
• \(A^c\) , see set, complement of .
• \(B(x,r)\) , see ball , 0NW .
• \( C^k\) , 2D0 .
• \(C\) , see function, continuous .
• \(ℂ \) , see complex numbers , 00B .
• \(C^0\) , see function, continuous .
• \(C^k\) , 1GZ .
• \( C_b\) , 0ZM , 14D .
• \(D(x,r)\) , see disk .
• \(\Delta \) , see set, symmetric difference , 23S .
• \(F_\sigma \) , see F-sigma .
• \( G_\delta \) , see G-delta .
• \( 𝕀 \) , see identity matrix .
• \(ℕ \) , see natural numbers , 00B .
• \(ℕ _\text {ZF}\) , 26K .
• \(ℚ \) , see rational numbers , 00B .
• \( \overline ℝ \) , see extended line .
• \( ℝ /2\pi \) , 0Y4 .
• \(ℝ \) , see real numbers , see also real line , 00B .
• \( S^1\) , see circle .
• \(S(x,r)\) , see sphere .
• \(T_2\) , see Hausdorff .
• \(ℤ \) , see integer numbers , 00B .
• \(\Leftrightarrow \) , 00D .
• \(\Vert \cdot \Vert \) , see also norm .
• \( \Vert \cdot \Vert _\infty \) , in \( C_b\) , 0ZM , 14D .
• \( \Vert \cdot \Vert _\infty \) , in \( ℝ ^n\) , 2CK and following , 10C .
• \( \Vert \cdot \Vert _p\) , in \( ℝ ^n\) , 2CK and following , 10C .
• \(\approx \) , 1NX , 1P1 .
• \( \underline \bigcap \) , 252 , 0J1 .
• \(\bigcap \) , 1W1 .
• \( \underline \bigcup \) , 026 , 25Z , 0KS , 0KZ .
• \(\bigcup \) , 1Y2 .
• \(\cap \) , 1W1 .
• \(\cup \) , 1Y2 , 026 .
• \(⧺ \) , see concatenation , 21W .
• \(\lci \) , 2FG .
• \(\lfloor x \rfloor \) , see floor .
• \(\loi \) , 2FG .
• \(\lv \) , 2FG , 071 .
• \(\neg \) , 00D .
• \(\rci \) , 2FG .
• \(\roi \) , 2FG .
• \(\rv \) , 2FG , 071 .
• \(\setminus \) , see set difference .
• \(\sim \) , 1NW , 1P1 , 1PN .
• \(\sim _f\) , 1PN .
• \(\subset \) , 1W0 .
• \(\subseteq \) , 1W0 .
• \(\subsetneq \) , 1W0 .
• \(\vee \) , 2DM , 00D .
• \(\wedge \) , 2DM , 00D .
• \( \searrow \) , 1HS .
• \( \partial f\) , see subdifferential .
• = , see equality .
• \(e\) , see Euler's number .
• sin(1/x) , 0RP .
• (partial) , 24W .
• Abel , 1KJ , 1T3 .
• Abel identity , 1T3 .
• Arzelà , 1HQ , 1K4 .
• Ascoli , 1HQ , 1K4 .
• Babylonian method , 0DN .
• Baire , 0VV .
• Baire category , 0VZ , 152 .
• Baire's, theorem , 0VV .
• Banach , see also theorem, Hahn--Banach — .
• Banach, space , 0ZM , 118 , 144 , 14D .
• Bessel , 1KM .
• Borel , 2CB , 0V3 .
• C , see function, continuous .
• CH, , see continuum hypothesis .
• Cantor, intersection theorem , 0BF , 0J6 , 0VP .
• Cantor, set , 2FB , 0XC , 0ZJ , 19Y , 0W4 .
• Cartan , 1M1 .
• Cartesian product , 2FG , 242 , 024 , 02H , 1WY , 05T , 05X , 0M3 , 0M5 , 0QM .
• Cartesian product, and topology , 2F7 .
• Cartesian product, of balls , 0QM .
• Cartesian product, of groups , 0X8 .
• Cauchy , 1QB .
• Cauchy, condensation test , 21D .
• Cauchy, product , 0FH , 1KQ .
• Cauchy, sequence , see sequence, Cauchy .
• Cohen , 2F2 .
• Darboux property , 1C8 .
• Darboux, example , 1CP .
• Dedekind , 04G , 04M .
• Dedekind-infinite , 04G , 04M .
• Descartes, rule of signs , 1D7 .
• Dini , 19S , 1HS .
• Dirichlet criterion , 21F .
• Dirichlet criterion, for integrals , 1TX .
• Dirichlet's approximation theorem , 0C1 .
• Edelstein , 0TH .
• Euclidean division , 28J .
• Euler , 1M3 , 1M7 .
• Euler's number , 1M3 .
• Euler, identity , 1M7 .
• Euler-Mascheroni constant , 0D6 .
• F-sigma , 0QC , 2CX , 16N , 16Q , 16S .
• F-sigma, \( ℝ \setminus ℚ \) , 152 .
• Faà Di Bruno , 1DJ .
• Fraenkel , 01J , 241 .
• Frobenious , 11H .
• G-delta , 0QC .
• Gamma function , 1BM .
• Gronwall , 1QB .
• Gödel , 2F2 .
• Hadamard , 1F1 .
• Hahn , see also theorem, Hahn--Banach — .
• Hamel basis , 02D , 057 .
• Hausdorff , 0G8 , 0J5 , 0J6 , 0J8 , 0P5 .
• Heine , 2CB , 0V3 .
• Hermite , 1F7 .
• Hilbert , 1P5 .
• Hoelder , 1J3 .
• Hospital , see Hôpital .
• Hurwitz , 1ZW .
• Hôpital rule , 1C5 .
• Hölder , 162 , 163 .
• Hölder inequality , 10M .
• \( 𝕀 \) , see identity matrix .
• Jacobi , 1G8 , 1V2 .
• Jacobi, formula , 1MT , 1V2 , 1V4 .
• Jacobi, matrix , 1G8 , 1GQ .
• Jensen inequality , 1C3 .
• Jordan , 2FW .
• Karush , 1HH .
• Koch , 0ZG .
• Kuhn , 1HH .
• Lagrange , 1H8 , 1HB .
• Lagrange multiplier , 10F , 10M , 1H8 , 1HB .
• Lagrange's theorem , 1C5 .
• Landau symbols , 1FB , 1FD , 1FF , 1FG , 1FJ .
• Laplace , 1V2 .
• Laplace expansion , 1V2 .
• Leibniz , 1DG .
• Leibniz's formula , 1DG .
• Leibniz, test , 0CN , 238 .
• Lindelöf , 1QB .
• Lipschitz , 162 , 163 , 1QB .
• Mazur , 2CH .
• Mertens , 0FM .
• Minkowski , 0RC , 0YJ .
• Minkowski sum , 0BC , 0RC , 2CP , 11R .
• Minkowski, dimension , 0YJ .
• Napier , 1M3 .
• Napier's constant , see Euler's number .
• Newton , 205 .
• ODE , 1QB and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• Osgood uniqueness condition , 1QR .
• Peano , 1XB , 1P5 .
• Picard , 1QB .
• Raabe , 0DR .
• Ricci , 0MM .
• Riemann , see also Riemann integral , see also Riemann integrable function .
• Riemann integral , 14T , 19K and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — 1FZ .
• S-saturated , 24X .
• \(T_2\) , see Hausdorff .
• Taylor , 1C5 .
• Taylor's theorem , 1C5 , 2D2 and following , — — — — — — 1FJ , — — — —
• Taylor's theorem, in \( ℝ ^n\) , 1G8 , 1GB .
• Taylor's theorem, with integral remainder , 1BR , 1FR .
• Taylor, series , 1C2 , 1KZ , 1NC .
• Tucker , 1HH .
• Tychonoff , 0MP .
• UC , see function, uniformly continuous .
• Ulam , 2CH .
• Von Neumann , 26D .
• Young inequality , 10M , 194 , 1V7 .
• ZF , see formal set theory , 2DX , 2F2 , 04M .
• ZFC , 2DX , 2F2 , 04M .
• Zermelo , 01J , 241 , 23R .
• Zorn , 23R , 17J .
• \( \Gamma \) , see Gamma function .
• \( \Vert \cdot \Vert _\infty \) , in \( ℝ ^n\) , — — — — — — — — — — —
• \( \Vert \cdot \Vert _p\) , in \( ℝ ^n\) , — — — — — — — — — — —
• accumulation point , 0HJ , 0HT , 0QP , 2F3 , 138 , 13W , 13Y .
• accumulation point, in a topological space , 0GY .
• accumulation point, in metric spaces , 2C2 and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — 0QN , — — — — — — — — — 0S8 , 0SB .
• accumulation point, in the real line , 0BG , 0B3 , 0BH , 0SD , 1JS .
• adherent point , 0QP , 0QP .
• adherent point, in a topological space , 0GX .
• adherent point, in metric space , 0NX .
• adjugate matrix , 1V2 .
• algebraic number , 0C7 .
• alternating series test , see Leibniz test .
• analytic function , 1N4 and following , — — — — — — —
• anti-discrete topology , see indiscrete topology .
• antireflexiv, relation , 23X , 224 , 24K .
• antisymmetric, relation , 23X , 224 , 24K .
• arc , 1NV .
• archimedean , 1ZK .
• associative, addition , 27Q .
• atom , 1YS , 00G , 01J .
• axiom, first --- of countability , 0MC , 13Y .
• axiom, of choice , 23R , 02H , 02H , 2GF .
• axiom, of empty set , 014 .
• axiom, of extensionality , 1Y8 , 241 .
• axiom, of foundation , 01R , 25G .
• axiom, of infinity , 241 , 01Y , 243 .
• axiom, of pairing , 1Y3 .
• axiom, of power set , 1Y1 , 022 .
• axiom, of regularity , 241 , 01R , 25G .
• axiom, of replacement , 241 .
• axiom, of specification , 1Y0 .
• axiom, of union , 1Y2 , 026 .
• axiom, second --- of countability , 0MD , 0MH , 0Q5 , 0Q7 , 0T4 .
• axioms, Peano's --- , 1XB .
• axioms, Zermelo–Fraenkel , see formal set theory .
• axioms, Zermelo—Fraenkel , 01J .
• ball , 0NW , 0P1 , 0SM , 106 .
• ball packing , 0YS .
• ball, in ultrametric , 0WR .
• ball, inclusion , 0NZ .
• base, (induction) , 1XC .
• base, (topology) , 0HR , 2B5 and following , 0KK , — — — — — — — — — — — — — — 2F5 , — — 0QJ .
• base, (vector spaces) , see basis .
• basis, (induction) , 1XC .
• basis, (topology) , see base .
• basis, (vector spaces) , 02D , 057 .
• belonging , 1X2 .
• biconditional , 00D .
• big O , see Landau symbols .
• binomial , see also theorem, binomial .
• binomial, coefficient , 205 , 1FP .
• binomial, series , 1FP .
• boundary , 0G7 , 0H7 , 0HM , 174 .
• boundary, repeatedly , 0H9 .
• bounded above , 22R .
• bounded below , 22R .
• bounded, totally , 0V3 , 0VG , 2GB .
• box , 0Z7 .
• cancellation , 27V , 28B , 28R .
• cardinality , 1YW and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• cardinality, comparison , 1YW .
• cardinality, continuum --- , 1YW , 0MZ .
• cardinality, countable , 1YW .
• cardinality, finite --- , 1YW , 1B1 .
• cardinality, of the continuum , 2F3 .
• cardinality, of the continuum , 03V .
• category , see also Baire's theorem , see also set, first/second — .
• category, Baire , 0VZ , 152 .
• chain , 01P .
• characteristic, function , 05R , 05S , 0BP , 2BP , 19Y .
• circle , 0Y3 , 0Y4 .
• closed , see set, closed — .
• closed curve , see curve, closed .
• closed simple curve , see curve, closed simple .
• closed topologist's sine curve , 0RP .
• closure , 0G7 , 0HM , 0KS , 0P8 , 0PP , 0PQ , 0Q8 , 176 , 178 , 0PN .
• closure, and interior , 0GJ , 0SG .
• closure, in metric space , 0NX .
• closure, repeated , 0GH .
• cluster point, in a metric space , 0QX , 0SN .
• codiscrete topology , see indiscrete topology .
• cofactor matrix , 1V2 .
• cofinal , 06P , 2B6 .
• commutative, group , 1ZF .
• commutative, ring— , 1ZG .
• compact set , 0J3 , 0J6 , 2CB , 0V3 , 0VP .
• compact set, and net , 0K8 .
• compact set, and ultrametric , 0X6 .
• compact, sequentially , 0V3 .
• comparable , 229 .
• complement, of a set , see set, complement of .
• complement, set , see set,complement .
• complete , 0X4 .
• complex numbers , 1ZD , 0FH , 19M , 1K9 , 1SN .
• concatenation , 21W .
• concrete topology , see indiscrete topology .
• conjunction , 00D .
• connected component , 0JT .
• connected set , 2BR .
• connection, in metric spaces , 2C5 and following , 0RR .
• constant, Euler-Mascheroni , 0D6 .
• continuity modulus , 156 , 15F , 161 , 163 , 1HR , 1J3 , 1JG .
• continuous function , 2B8 and following , 2B9 , 2DP and following , — — — — — — — — — — — — —
• continuum hypothesis , 2F2 , 2F3 .
• continuum, cardinality of the — , 03V , 2F3 .
• contrapositive , 00N .
• convergence, of a sequence , 0MT .
• convergence, of a series , 0CN and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — 0DW , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• convergence, pointwise --- , 0MP , 1HQ and following , 2DT , — — — — — — — — — — — — — — — — —
• convergence, total --- , 116 , 118 , 1T1 .
• convergence, uniform --- , 1HQ and following , 2DT , — — — — — — — — — — — — — — — — —
• convex combination , 16W .
• convex envelope , 130 , 2G4 .
• convex function , 0ZX , 16V and following , — — — — — — — — — — — — — — — — — — — — — — — — — 17Y , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — 1J3 .
• convex hull , see convex envelope .
• convex set , 2F0 and following , — — 16X , —
• convex set, strictly --- , 19D .
• countable , 04J , 2DD .
• countably infinite , 2DD .
• counterimage , 091 .
• criterion, total convergence --- , 118 .
• curve , 1NT and following , — 1NV , — — — — — — — — — — — — — — — — — — — —
• curve, Hilbert — , 1P5 .
• curve, Koch , 0ZG .
• curve, Peano — , 1P5 .
• curve, closed , 1PB .
• curve, embedded , 1NV , 1P0 .
• curve, immersed --- , 1NV .
• curve, parametric — , 1NV , 1PB .
• curve, polygonal — , 2G6 , 2FN .
• curve, simple , 1NV .
• curve, simple closed , 2FW , 2G6 , 1PB .
• curve, trace , 1NV .
• decreasing , 2DJ .
• deleted neighborhood , 0B2 , 0BG , 20D .
• deleted neighborhood, in a topological space , 0GY .
• dense , see set, dense .
• dense, in metric space , 0NX .
• derivative , 0HM .
• derivative, partial --- , 2D3 and following , — — — — — — —
• derivative, total --- , 2D3 and following , — — — — — — —
• determinant , see matrix, determinant .
• diffeomorphism , 1NX .
• differential , 2D3 and following , — — — — — — —
• differential equation , see ODE .
• dilation , 124 .
• dimension, Minkowski , 0YJ .
• dimension, box --- , 0Z7 .
• directed set , see order, directed , 2B2 .
• disconnected set , 2BR .
• discrete topology , 2F6 , 2FD , 2F9 , 0QF .
• discrete, distance , 2C1 .
• discrete, topology , 2C1 .
• disjunction , 00D .
• disk , 0NW , 0P1 , 0PY , 0Q0 , 0SM , 0VD , 106 .
• disk, in ultrametric , 0WR .
• distance , 0MS .
• distance function , 2C4 and following , 0R8 .
• distance function, and convex sets , 198 , 19B .
• distance, discrete --- , 2C1 .
• distance, p-adic --- , 0XF .
• \(e\) , see Euler's number .
• ear , 0JN .
• embedded curve , see curve, embedded --- .
• empty set , 242 , 014 .
• enumeration , 2DF , 1T9 .
• epigraph , 13D , 181 .
• equality , 1YS .
• equality, in set theory , 1Y8 .
• equations, Bessel's --- , 1KM .
• equicontinuous family , 1HR , 1JG .
• equicontinuous functions , 1J3 .
• equinumerous , 22B .
• equipotent , 22B .
• equivalence relation , see relation, equivalence .
• equivalent, norms , 107 , 109 .
• erosion , 124 .
• evaluation, of well-formed formula , 00J , 00N .
• eventually , 02M , 06Y , 064 , 018 , 0B3 , 20D , 20G , 20N , 0DJ , 02F , 21B , 21C , 0DR , 2B6 .
• exchanging limits , 0CX .
• expansion, Taylor's --- , see Taylor's theorem .
• exponential , 1M3 , 1SD .
• exponential, matrix --- , 1MF .
• exponentiation , 228 .
• exponentiation, in a field , 202 .
• exponentiation, of natural numbers , 280 .
• extended line , 09X , 0HP .
• fattened set , 0RC .
• field , 1ZH .
• field, ordered — , 1ZX .
• field, ordered —, \(ℂ \) , 08V .
• filtering property , see order, with filtering property .
• finite , see set, finite .
• finite linear combination , 02D .
• first axiom of countability , 0MC , 13Y .
• first category set , 0VW .
• fixed point , 16J , 16K .
• floor , 0BS , 0BY , 0D6 .
• formal set theory , 01J , 241 and following , — — — — — — — — — — — — — — — — — — — — — — — — — —
• formula, Leibniz's --- , see Leibniz's formula .
• formula, Taylor's --- , see Taylor's theorem .
• formula, atomic , 00G .
• formula, exists and is unique , 013 .
• formula, well-formed — , 1YS , 1YS , 00G , 1YK , 00K , 00R , 013 , 252 , 05Z .
• formula, well-formed —, evaluation , 00J , 00N .
• formula, well-formed —, in set theory , 01J .
• formula, well-formed —, with quantifiers , 00Q , 00V .
• fractional part , 0BT .
• free variable , 00Q .
• frequently , 06Y , 064 , 018 , 0B3 .
• function , 1Y6 .
• function, Gamma , 1BM .
• function, Hölder --- , 2DR and following , 162 , — — 163 , — — — — — — — 16F , — — — 1GX , 1J3 .
• function, Lipschitz --- , 0NH , 0R9 , 0Y0 , 156 , 2DR and following , 162 , — — 163 , — — — — — — — 16F , — — — 17W , 18J , 1GX , 1GZ .
• function, Riemann integrable --- , 14T , 19K and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — 1J3 .
• function, absolutely homogeneous , 0ZV .
• function, analytic , see analytic function .
• function, bi--Lipschitz , 0Y0 .
• function, bounded --- , 142 , 1J3 .
• function, characteristic , 05R , 05S , 0BP , 2BP , 19Y .
• function, continuous , see continuous function .
• function, continuous --- , 2DN .
• function, convex , see convex function .
• function, discontinuous , 152 .
• function, indicator , see characteristic function .
• function, left inverse , 2BZ .
• function, liminf of — , see liminf .
• function, limsup of — , see limsup .
• function, monotonic , 1J3 .
• function, monotonic --- , 147 .
• function, partial , 23X , 01P .
• function, piecewise constant --- , 144 .
• function, piecewise smooth --- , 1QK .
• function, positively homogeneous , 0ZW , 0ZX .
• function, proper --- , 0T3 .
• function, regulated , see regulated function .
• function, right continuous --- , 147 .
• function, semi continuous , see upper/lower semicontinuous .
• function, strictly convex , see strictly convex function .
• function, uniformly continuous --- , 2DQ and following , — 155 , — — — — 15F , — — — — — — — — 163 , 16F , 1J3 , 1JN , 1JQ .
• function, uniformly continuous, space of --- , 1JX .
• functional, relation , 23X , 1Y6 .
• functions, equicontinuous , 1J3 .
• fundamental system of neighbourhoods , 0GW , 0MM , 2BP .
• funzione distanza , 0R9 .
• generate , 02D , 057 .
• graph , 1Y6 .
• greatest common divisor , 1WH .
• greatest element , 229 .
• greatest lower bound , see infimum .
• group , 1ZF .
• homeomorphism , 0J8 , 2B9 , 09T , 2F3 , 0V8 , 21N , 1NW , 1P0 , 1PH , 1PN .
• hyperplane , 17J .
• identity matrix , 1MG .
• image , 092 .
• immersed curve , see curve, immersed .
• implication , 00D .
• incomparable , 229 .
• increasing , 2DJ .
• indicator , see characteristic function .
• indiscrete topology , 2F6 .
• induced norm , 2CM .
• induction , 1XC .
• induction principle , 23B , 1XC .
• induction principle, strong -- , 1XS .
• induction, strong , 1XS .
• induction, transfinite --- , 1XY .
• inductive , see S-saturated .
• inequality, Jensen — , 1C3 .
• inequality, Young — , see Young inequality .
• inequality, triangle --- , 0MS , see triangle inequality .
• inf , see infimum .
• inf-convolution , 13P .
• infimum , 22R , 209 , 20B .
• infinite , see set, infinite .
• infinite, countably --- , 2DD .
• informal set theory , 01J .
• initial segment , 07T .
• injective , 23X .
• integer numbers , 03M .
• integer numbers, dense in ultrametric , 0XW .
• integer part , see floor , 0BY , 0D6 .
• integral domain , 203 .
• interior , 0G7 , 0GF , 0GH , 0HM , 0KS , 0P3 , 0PB , 0PD , 0PG , 0PJ , 0PM , 0Q8 , 172 , 174 , 176 , 178 .
• interior, and closure , 0GJ , 0SG .
• interior, in metric space , 0NX .
• interpolation, polynomial --- , 09J , 1F7 .
• intersection theorem , see Cantor, intersection theorem .
• interval , 2DW and following , 07C , — — — — — — — —
• interval, standard — , 07D .
• irrational numbers , 0C1 , 0VZ , 152 , 19Y .
• irrational numbers, approximation , 29Q .
• irreflexiv, relation , 23X , 224 , 24K .
• isolated point , 0HD , 0HJ , 2F3 , 0T5 , 1TZ .
• isolated point, in a topological space , 0GX .
• isometry , 0TM , 0TT , 0TW , 0TZ .
• itersection of sets , 1W1 .
• l.s.c. , see lower semicontinuous .
• labeled polygon , 2CD .
• least element , 229 .
• least upper bound , see supremum , 08T .
• left inverse , 2BZ , 2BX .
• lemma, Abel's — , 1KJ .
• lemma, Dini's --- , 1HS .
• lemma, Gronwall's — , 1QB .
• lemma, Hadamard's — , 1F1 .
• lemma, Zorn's — , 23R , 17J .
• lexicographic order , see order, lexicographic .
• liminf , 29P , 02F .
• liminf, of function , 29P and following , — 20F , — — 0BK , — — — — — — — —
• liminf, of sequence , 0BQ .
• liminf, of sets , 1Z2 , 063 , 064 , 065 , 0BP .
• limit inferior , see liminf .
• limit point, of a net in a topological space , 2B4 .
• limit superior , see limsup .
• limsup , 29P .
• limsup, of function , 29P and following , — 20F , — — 0BK , — — — — — — — —
• limsup, of sequence , 0BQ .
• limsup, of sets , 1Z2 , 063 , 064 , 065 , 0BP .
• line , see also real line .
• line, extended , see extended line .
• linear isometry , 111 .
• linear, order , see order, total .
• linearly independent , 02D , 057 .
• locally compact , 0VD .
• logarithm , 21N .
• lower bounds , 22R .
• lower semicontinuous , 2CV and following , 138 , — — — — — — — — — — — — —
• majorants , 22R .
• matrix, adjugate --- , 1V2 .
• matrix, cofactor --- , 1V2 .
• matrix, determinant , 1V0 , 1V2 , 1V4 .
• matrix, exponential , 1MF .
• matrix, identity --- , 1MG .
• maximal , 229 , 06V .
• maximum , 229 , 06V , 0HT .
• mean value theorem , see Lagrange's theorem .
• metric space , 0MR and following , 2CC , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — 13P .
• metric space, also a group , 0R5 .
• minimal , 229 , 1XP .
• minimum , 229 .
• minimum, on convex set , 18B .
• minorants , 22R .
• monotonic , 2DJ .
• natural numbers , 1X9 and following .
• natural numbers, order , 26Y .
• negation , 00D .
• neighborhood, deleted , 0B2 , 0BG , 20D .
• neighborhood, deleted, in a topological space , 0GY .
• neighborhood, of infinity , 231 .
• neighborhood, punctured , 0B2 .
• neighbourhood , 0GW .
• neighbourhood, fundamental system of — , 0GW , 0MM , 2BP .
• neighbourhood, in \( ℝ \) , 0B2 .
• net , 21J , 230 , 22Y , 0K4 , 0KD , 0KG , 2BP .
• net, and compact set , 0K8 .
• network , 0FS .
• norm , 0ZV .
• norm, Frobenious , 11H .
• norm, and dimension , 0Z1 .
• norm, spectral , 11G .
• normed vector space , 0ZT and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• normed vector space, strictly convex --- , 0ZZ .
• norms, equivalent , 107 , 109 .
• numbers , see also natural numbers , see also integer numbers , see also rational numbers , see also real numbers , see also complex numbers .
• numerabile , 2DD .
• numero di Nepero , see Euler's number .
• one-point compactified line , 0HR .
• open, see{set, open —} , 0G6 .
• open-close , 0Q8 .
• order , 1YY and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• order isomorphism , 07V .
• order relation , 1Y5 .
• order topology , 2F5 , 2F9 .
• order, (partial) , 24W .
• order, directed , 2FJ and following , — 06N , — — — — — — — — — — — — — — — 237 , 0HS , 0HT , 0HW , 0K5 .
• order, directed, of sets , 0GQ , 0KG .
• order, lexicographic , 2FH and following , — 071 , — — — — — 2F9 .
• order, of natural numbers , 26Y .
• order, partial , 1Y5 .
• order, strict total --- , 24K .
• order, total , 1Y5 , 24W , 24K , 2F5 , 2F9 .
• order, type , 07V .
• order, with filtering property , 2FJ and following , — 06M , — — 06P , — — 06Q , — — — — — — — — — — — 237 , 2B6 .
• order, with filtering property, and net , 21J .
• order-isomorphic , 07V .
• ordered field , 1ZX .
• ordered field, \(ℂ \) , 08V .
• ordered ring , 1ZJ , 1ZT , 1ZV .
• ordered set , see order .
• ordinal , 26D .
• ordinary differential equation , see ODE .
• oscillation , 13T , 143 .
• osculating circle , 1TJ , 1TJ .
• p-adic, distance , 0XF .
• p-adic, valuation , 0XG , 0XM .
• parametric curve , see curve, parametric --- .
• partial function , 23X , 01P .
• partial order , see order, partial , 224 .
• partial, derivative , see derivative, partial .
• path connected set , 0RG .
• perfect , 09S , 2F3 , 0W3 .
• pointwise , see convergence, pointwise .
• polygon , 2FN , 10Y .
• polygon, ear , 0JN .
• polygonal curve , 2G6 , 2FN .
• polynomial , 09J , 03H , 29Q and following , — — — — — — — — — 0C7 , — — — — — — — — — 19M , 1F7 , 23Z , 1SC , 1SD , 1SF , 1SH , 1SK , 1SN , 1SP .
• polynomial interpolation , 09J , 1F7 .
• polynomial, Taylor's --- , see Taylor's theorem .
• polynomial, convergence of --- , 1J3 .
• polynomial, ring --- , 0C8 .
• polynomial, sequence of --- , 1J3 .
• power , see also exponentiation .
• power series , 1K6 and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• power set , 242 , 1Y1 .
• predecessor , 25C , 1Z1 , 1YP .
• preorder , 02M , 1Z7 .
• product topology , 0M3 , 0QM .
• product topology (of infinitely many spaces) , 2F7 , 2F9 .
• product, Cartesian — , see Cartesian product .
• projection, theorem , 17D .
• proper, function , 0T3 .
• property, cancellation , see cancellation .
• proposition (logic) , 1YS .
• punctured neighborhood , 0B2 .
• quantified variable , 00Q .
• radius of curvature , 1TJ .
• ratio test , 21C .
• rational numbers , 03M , 1ZD , 200 , 29Q , 0C1 , 19Y , 1T9 .
• rational numbers, and ultrametric , 0XF , 0XM , 0XY , 0Y0 .
• real line , see also real numbers .
• real line, one-point compactified --- , 0HR .
• real number, approximation of — , 29Q .
• real numbers , 1ZD , 09X and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — see also real line .
• recursive , 0CN .
• recursive, definition , 08Z .
• reflexiv, relation , 23X .
• regular curve , see curve, immersed .
• regulated function , 2CT and following , 141 , 142 , 143 , 144 , 1B3 , 1B4 , 1B6 , 1B9 , 1J3 .
• relation , 1WY .
• relation, antireflexiv , 224 .
• relation, antisymmetric , 224 .
• relation, equivalence --- , 1W5 , 093 , 0MX , 0MZ , 0R2 , 0YD .
• relation, equivalence ---, between curves , 1NW , 1NX , 1PN .
• relation, equivalence ---, for \( S^1\) , 0Y4 .
• relation, equivalence ---, in group , 0R5 .
• relation, irreflexiv , 224 .
• relation, order — , see order .
• relation, transitive , 224 .
• right inverse , 2BX .
• rigidity property , 1JG .
• ring , 1ZG .
• ring, of polynomials , 0C8 .
• ring, ordered — , 1ZJ , 1ZT , 1ZV .
• root test , 219 .
• rule of signs , 1D7 .
• second axiom of countability , 0MD , 0MH , 0Q5 , 0Q7 , 0T4 .
• second category set , 0VW .
• semicontinuous, lower , see lower semicontinuous .
• semicontinuous, upper , see upper semicontinuous .
• separable space , 0MH , 0Q7 .
• separation , 17H , 17J , 17T .
• sequence , 16G .
• sequence, Cauchy — , 0MT , 0MV , 0N5 , 0N6 , 0N8 , 0NC , 0VD .
• sequence, Cauchy —, and subsequence , 0N8 , 0NC .
• sequence, convergence of --- , see convergence of a sequence .
• sequence, recursive , 08Z , 0CN .
• sequentially compact , 0V3 .
• series, binomial — , 1FP .
• set theory , see also axioms ... .
• set theory, formal , 01J .
• set theory, informal , 01J .
• set, Cantor -- , see Cantor set .
• set, Dedekind—infinite , 04G , 04M .
• set, boundary , see boundary , 0H7 .
• set, closed —, in metric space , 0NX .
• set, closed —, in topology , 0G6 .
• set, closure , see closure .
• set, complement of a --- , 23S , 05R , 063 , 0G5 , 0H7 , 0P6 .
• set, countable --- , 04J .
• set, dense , 0G7 .
• set, derived --- , 0GY , 2C2 , 0QV , 0SD , 0T5 .
• set, difference , 23S .
• set, empty — , 242 , 014 .
• set, fattened --- , 0RC .
• set, finite --- , 1B1 .
• set, first category --- , 0VW .
• set, infinite --- , 1B1 .
• set, infinite, Dedekind --- , 04G , 04M .
• set, interior , see interior .
• set, of finite subsets , 03R , 053 , 0F5 , 0FW .
• set, open —, in metric space , 0NX .
• set, open —, in topology , 0G6 .
• set, path connected , 0RG .
• set, perfect --- , 09S , 2F3 , 0W3 .
• set, power -- , see power set .
• set, power — , 242 .
• set, second category --- , 0VW .
• set, strongly directed , see order, with filtering property .
• set, sublevel --- , 182 .
• set, symmetric difference , 23S .
• set, symmetric difference, using characteristics , 05S .
• simple closed curve , see curve, closed simple .
• simple curve , see curve, simple --- .
• simplex , 16Z .
• sin(1/x) , 0RP .
• sine curve , 0RP .
• small o , see Landau symbols .
• snowflake , see Koch curve .
• space, Hausdorff --- , see Hausdorff .
• space, of uniformly continuous functions , 1JX .
• space, separable , 0MH .
• space, topological , 0G5 and following , 2DY , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• space, totally disconnected --- , 0K0 , 0QF , 0WW .
• span , 02D , 057 .
• sphere , 0PY , 0Q0 , 0SM , 106 , 114 , 1P7 , 1PX .
• star , see Koch curve .
• strict partial order , 224 .
• strict total order , 24K .
• strictly convex function , 17Y .
• strictly convex, normed vector space , 0ZZ .
• strong induction , 1XS .
• strongly directed set , see order, with filtering property .
• subadditive function , 0N1 , 15W , 192 .
• subadditive function and ultrametric , 0WY .
• subdifferential , 188 , 19B .
• sublevel set , 182 .
• subnet , 230 , 22Y .
• subscript , 228 .
• subsequence , 0D2 , 0D4 , 230 .
• subsequence, converging , 0MP , 0V3 .
• successor, in Peano's natural numbers , 1XB .
• successor, in Zermelo—Fraenkel set theory , 24X , 1YM , 26F .
• successor, in well ordered sets , 1Z0 .
• sum, Minkowski --- , 0RC .
• sup , see supremum , 08T .
• superscript , 228 .
• support , 17T .
• support, of a curve , 1NV .
• supporting hyperplane , 17J .
• supremum , 22R , 08T .
• surjective , 23X .
• symmetric, relation , 23X .
• tautology , 00N , 016 , 05Z .
• tessellation , 0Z7 .
• test, Cauchy condensation --- , 21D .
• test, Leibniz — , 0CN , 238 .
• test, alternating series — , see Leibniz test .
• test, ratio --- , 21C .
• test, root --- , 219 .
• theorem, Ascoli--Arzelà's --- , 1HQ , 1K4 .
• theorem, Baire's — , 0VV .
• theorem, Cauchy–Lipschitz — , 1QB .
• theorem, Dirichlet's approximation , 0C1 .
• theorem, Edelstein’s --- , 0TH .
• theorem, Hahn--Banach — , 17J .
• theorem, Hurwitz's --- , 1ZW .
• theorem, Hôpital , see Hôpital rule .
• theorem, Jordan — , 2FW .
• theorem, Lagrange's --- , see Lagrange's theorem .
• theorem, Mazur–Ulam , 2CH , 114 .
• theorem, Mertens' --- , 0FM .
• theorem, Monotone convergence --- , 0G0 .
• theorem, Picard–Lindelöf — , 1QB .
• theorem, Taylor's --- , see Taylor's theorem .
• theorem, Taylor's ---, with Lagrange remainder , see Lagrange remainder .
• theorem, Tychonoff — , 0MP .
• theorem, Zermelo , 23R .
• theorem, binomial --- , 205 .
• theorem, dimension --- , 057 .
• theorem, existence and uniqueness — , 1QB .
• theorem, implicit function --- , 1GD .
• theorem, intersection --- , see Cantor, intersection theorem .
• theorem, mean value --- , see Lagrange's theorem .
• theorem, of uniqueness of the limit , 0MW .
• theorem, projection --- , 17D .
• theorem, two ears — , 0JN .
• topological group , 0X8 .
• topological space , 0G5 and following , 2DY , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• topology , 2DY , 0G6 .
• topology, discrete , 2F6 , 2FD , 2F9 , 0QF .
• topology, discrete --- , 2C1 .
• topology, in metric spaces , 2C2 and following , — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
• topology, indiscrete , 2F6 .
• topology, induced — , 2BR , 2DK .
• topology, order — , 2F5 , 2F9 .
• topology, product — , 0M3 , 0QM .
• topology, product — (of infinitely many spaces) , 2F7 , 2F9 .
• total convergence , see convergence, total .
• total convergence criterion , 118 .
• total order , see order, total .
• total, derivative , see derivative, total .
• total, relation , 23X , 1Y6 .
• totally bounded , 0V3 , 0VG , 2GB .
• totally disconnected , see space, totally disconnected .
• trace, of a curve , 1NV .
• transcendental number , 0C7 .
• transfinite, induction , 1XY .
• transitive, relation , 23X , 224 , 24K .
• transitive, set , 24Z .
• triangle inequality , 0B0 , 0MS , 0WN , 0ZV , 0ZX .
• triangulated , 1XW .
• trichotomous, relation , 23X , 24K .
• trivial topology , see indiscrete topology .
• u.s.c. , see lower semicontinuous .
• ultrametric , 0WM .
• ultrametric, of sequences , 0X0 .
• ultrametric, of sequences, dimension , 0ZP .
• uniform , see convergence, uniform .
• union of sets , 1Y2 , 026 .
• unlabeled polygon , 2CD .
• upper bounds , 22R .
• upper semicontinuous , 2CV and following , 138 , — — — — — — — — — — — — —
• valuation, p-adic --- , 0XG , 0XM .
• variable, free , 00Q .
• variable, quantified , 00Q .
• vector space , 0MM , 1SN .
• vector space, normed --- , see normed ... .
• well-formed formula , see formula, well-formed .