text stringlengths 71 10k |
|---|
ge of a subspace of the Cantor Space (G, τ ). Further, if (X, τ 1) is compact, then the subspace can be chosen to be closed in (G, τ ). Let φ be the continuous mapping of (G, τ ) onto I∞ shown to exist in the Proof. proof of Theorem 9.3.8. By Urysohn’s Theorem 9.4.19, (X, τ 1) is homeomorphic to a subspace (Y, τ 2) of ... |
ductive definition of the sequence of points and prove (i) the sequence of points tends to a limit L(a) in S; (ii) if λ ∈ [0, 1] is represented by distinct binary decimals a, a then L(a) = L(a); hence, the point L(λ) in S is uniquely defined; (iii) if f : [0, 1] → S is given by f (λ) = L(λ), then f is surjective; (iv) f ... |
d for i ∈ I, let Ci be a closed i∈I Ci is a closed subset of i∈I (Xi, τ i). 10.1.4 Proposition. a family of topological spaces having product space ( i ∈ I, Bi is a basis for τ i, then Let I be a set and and let {(Xi, τ i) : i ∈ I} be If for each i∈I Xi, τ ). Oi : Oi ∈ Bi and Oi = Xi for all but a finite number of i B =... |
) 10.2.8 Remark. We saw in the order diagram above that the elements d and c are not comparable. Also 1 and e are not comparable. In N, Q, R, and Z with the usual orderings, every two elements are comparable. In Example 10.2.4, 3 and 5 are not comparable. 10.2.9 Definitions. A partially ordered set (X, ) is said to be ... |
gical spaces are homeomorphic to subspaces of cubes? We now address this question. Let (X, τ ) be a topological space. Then (X, τ ) is 10.3.8 Definitions. said to be completely regular if for each x ∈ X and each open set U x, there exists a continuous function f : (X, τ ) −→ [0, 1] such that f (x) = 0 and If (X, τ ) is ... |
= V1 \ U 1, U Then we inductively define 1 ∩ B = U1 ∩ B, U n = Un \ n V i and V n = Vn \ n U i and V 1 ∩ A = V1 ∩ A. i=1 n ∩ B = Un ∩ B, i=1 n ∩ A = An ∩ A. and V So that U n ∈ τ , U n=1 U n ∈ τ , V Now put U = ∞ Then U ∩ V = Ø, U ∈ τ , Hence (X, τ ) is a normal space. n and V = ∞ n=1 V n. V ∈ τ , A ⊆ V , and B ⊆ U . W... |
literature there are alternative proofs using uniform continuity. 10.3. TYCHONOFF’S THEOREM 291 Noting the definitions of Sr and Ts and using (1) and (2), we can apply Remark , to show that there exists a 10.3.21, with A = Srn ∪ j∈J closed set Hn in (X, τ ) such that Hj and C = Tsn ∩ k∈K H0 k Srn ∪ j∈J Hj ⊆ H0 n ⊆ Hn ⊆... |
t a Lindelöf space. [Now we know from (ii) and (iv) that a product of two Lindelöf spaces is not necessarily a Lindelöf space.] (v) Verify that a topological space is compact if and only if it is a countably compact Lindelöf space. (See Exercises 7.2 #17.) 2. Prove that any product of regular spaces is a regular space.... |
over of X; that Ai = X. Then {Bj : j ∈ J}, J some index set, is said is each Ai ⊆ X and i∈I to be a refinement of the cover {Ai : i ∈ I} if {Bj : j ∈ J} is a cover of X and for each j ∈ J , there exists an i ∈ I such that Bj ⊆ Ai. (i) Prove that every cover {Ai : i ∈ I} of X is is also a refinement of itself. (ii) Prove ... |
S ⊆ X and S is dense in X}. The spread of (X, τ ) is the cardinal number s(X) given by s(X) = ℵ0 + sup{card (D) : D is a discrete subspace of X}. The Lindelöf degree, L(X), of (X, τ ) is the smallest infinite cardinal number ℵ such that every open cover of X has a subcover of cardinality ℵ. Of course L(X) = ℵ0 if and on... |
a vector subspace of B, and f : E → R a bounded linear functional. Then for any x ∈ B \ E, there exists a linear functional f1 : E1 = span{E, x1} → R that extends f (that is, f (x) = f1(x), for all x ∈ E) and satisfies ||f ||op = ||f1||op. (If X is a subset of a vector space L, span(L) denotes the smallest vector subspa... |
hen the infinite-dimensional Banach space B is reflexive. (For other special cases, see Argyros et al. [14].) To verify the reflexivity special case, check that the following statements are true: (a) Every infinite-dimensional Banach space has a subspace which is an infinite-dimensional separable Banach space. (b) Every clo... |
s a closed subspace of a compact Hausdorff space. Let φ be any continuous mapping of (X, τ ) into any compact Hausdorff space (Y, τ ). We are required to find a mapping Φ as in Definition 10.4.1 so that the diagram there commutes and show that φ is unique. Let F(Y ) be the family of all continuous mappings of (Y, τ ) into ... |
]. So card I = cℵ0 = c. Then card ([0, 1]I ) = cc = 2c. Thus card βN 2c. Hence card βN = 2c. The proposition then follows from Proposition 10.4.17. Let X be any unbounded subset of Rn, for any 10.4.18 Proposition. If τ is the euclidean subspace topology on X, then βX, the Stonen ∈ N. ˘Cech compactification of (X, τ ), h... |
map.) 5. Noting the definition of S1 in Exercises 6.1 #15, let f : R → S1 be given by f (x) = (cos 2πx, sin 2πx). Show that f is a quotient mapping but not a closed mapping. Is f an open mapping? 6. Find an example of a quotient mapping which is neither an open mapping nor a closed mapping. 7. Show that every compact H... |
through the origin, but excuding the origin, in R3. Each line is of course determined by a non-zero vector in R3, unique up to scalar multiplication. RP2 is then the quotient space of R3 \ {0} under the equivalence relation v ∼ λv, for all λ ∈ R, λ = 0. (See also Definitions A5.0.1 and the following discussion.) In fac... |
ns from [a, b] into R. If there exist polynomials Pg and Ph such that for all x ∈ [a, b], |g(x)−Pg(x)| < ε 2 2, then putting P (x) = Pg(x) + iPh(x) we have that the and |h(x) − Ph(x)| < ε polynomial P satisfies |f (x) − P (x)| < ε. So to prove this theorem, it suffices to prove it for the special case: f is a continuous f... |
of a smooth function which is not analytic is given in Exercises 12.1 #10. 12.1.16 Theorem. of zeros of f ; that is, Z = {x : x ∈ R such that f (x) = 0}. Let f be a function of R into itself and Z be the set (i) If f is a non-constant polynomial, then Z is a finite set. (ii) If f is a non-constant analytic function, the... |
for any a, b ∈ [0, 1] with a = b, the Bernstein polynomial B1(f ), where f (x) = x, for all x ∈ [0, 1], is readily seen to satisfy B1(f ) = f . So (B1(f ))(a) = (B1(f ))(b). (c) On the other hand, the set {fn : fn(x) = sin(2πnx), x ∈ [0, 1], n ∈ N} of functions of [0, 1] into R does not separate the points 0 and 1, si... |
for every t ∈ [a, b]. Define the polynomial p by p(t) = p(t) − p(0), for all t ∈ [a, b], and note that |p(0)| < ε | |t| − p(t) | = | |t| − p(t) + p(0) | | |t| − p(t) | + |p(0)| < ε, for every t ∈ [a, b]. 2 . So we have p(0) = 0 and Since A is an algebra, the function p(f ) ∈ C(X, R), given by (p(f ))(x) = p(f (x)) for ... |
omorphic as a C∗algebra to B if there is a surjective one-to-one mapping φ : A → B such that φ(x + y) = φ(x) + φ(y), φ(αx) = αφ(x), φ(x · y) = φ(x) · φ(y), and φ(x∗) = (φ(x))∗, for all x, y ∈ A and α ∈ C. The Gelfand-Naimark Representation Theorem, proved using the Stone-Weierstrass Theorem, says every commutative unit... |
an immediate consequence of Proposition A1.1.11 and Example A1.1.8 we have the following result. A1.1.12 Corollary. Every subset of Z is countable. A1.1.13 Lemma. If S1, S2, . . . , Sn, . . . is a countably infinite family of countably infinite sets such that Si ∩ Sj = Ø for i = j, then infinite set. ∞ i=1 Si is a countab... |
ts. To explain what we mean by “bigger” we will need the next theorem. Our exposition is based on that in the book, Halmos [168] 405 A1.2.1 Theorem. (Cantor-Schröder-Bernstein) Let S and T be sets. If S is equipotent to a subset of T and T is equipotent to a subset of S, then S is equipotent to T . Proof. Without loss ... |
nto A is denoted by AB. Further, α β is defined to be card AB. Once again we need to check that the definition makes sense, that is that αβ does not depend on the choice of the sets A and B. We also check that if n and m are finite cardinal numbers, A is a set with n elements and B is a set with m elements, then there are... |
ch that x < a is said to be the initial segment of (S, ) determined by a. A1.4.17 Remark. Every initial segment of an ordinal number is an ordinal number. A1.4.18 Proposition. For any ordinal numbers α and β, precisely one of the following is true: (i) α = β; (ii) α is an initial segment of β; (iii) β is an initial seg... |
pleted.” In 428 APPENDIX 2: TOPOLOGY PERSONALITIES 1939, just before the start of World War II, Banach was elected President of the Polish Mathematical Society. The Nazi occupation of Lvov in June 1941 meant that Banach lived under very difficult conditions. Towards the end of 1941 Banach worked feeding lice in a German ... |
a “space-filling curve”. The length of the curve is infinity, while the area enclosed by it is 5/12 that of the square. Two fractals – Sierpiński triangle and Sierpiński carpet – are named after him. Sierpiński continued to collaborate with Luzin on investigations of analytic and projective sets. Sierpiński was also hig... |
even integer, what can you say about the orbit of m? A3.3 Phase Portraits, Attracting and Repelling Fixed Points We wish to study dynamical systems, that is processes in motion. Such processes include for example the motion of planets, but other systems to which this theory is applied include the weather and populatio... |
ction. But the surprising feature is that the dynamics of Qc changes as c changes. The following theorem indicates this. We leave the proof of the theorem as an exercise. 452 APPENDIX 3: CHAOS THEORY AND DYNAMICAL SYSTEMS A3.5.6 Theorem. quadratic function for c ∈ R. (The First Bifurcation Theorem) Let Qc be the (i) If... |
(A2) = A1. (v) Observe that A2 ⊆ A1 ⊆ I1 and f 2(A2) = I1. (vi) Use mathematical induction to show that for n 3 there are closed intervals A1, A2, . . . , An−2 such that An−2 ⊆ An−3 ⊆ · · · ⊆ A2 ⊆ A1 ⊆ I1 such that f (Ai) = Ai−1, i = 2, . . . , n − 2, and f (A1) = I1. (vii) Deduce from (vi) that f n−2(An−2) = I1 and A... |
t this condition is automatically true if the two conditions in Definition A3.7.7 hold. Their work appeared in the paper “On Devaney’s definition of chaos" by the authors John Banks, Gary Davis, Peter Stacey, Jeff Brooks and Grant Cairns in the American Mathematical Monthly (Banks et al. [31]). See also (Banks et al. [32]... |
hat Y ⊆ Ui and, for each i ∈ I, diam Ui < ε. Then {Ui : i ∈ I} is said to i∈I be an ε-covering of the set Y . 471 We are particularly interested in ε-coverings which are countable. So we are led to ask: which subsets of a metric space have countable ε-coverings for all ε > 0? The next Proposition provides the answer. L... |
0 < xi < 1, i = 1, . . . , n}, prove that dimH S1 = dimH Rn. (iii)* Using the method of Proposition A4.1.24, show that if n = 2 then H2(S1) 2 and so dimH(S1) 2. (iv) Prove that dimH R2 2. (v)* Using an analogous argument, prove that dimH Rn n, for all n > 2. 6. Prove Proposition A4.1.19. [Hint. Prove that as.Hs(X) Hs(... |
problems dating back 2,000 years and played a key role in the progress of computer algebra. Now we set the stage for the Erlangen Program of Felix Klein. We have all met Euclidean geometry which has points, lines, angles, and a metric (distance) and of course the famous Pythagoras theorem for right angled triangles wh... |
: (i) G = N H and N ∩ H = {1}. (ii) Every element of G can be written uniquely as a product of an element of N and an element of H. If one (= both) of these is true, then G is called a semidirect product of N and H, written G = NH. Semidirect product is more general than product; if H is also a normal subgroup, then N ... |
oup of any abelian group is an abelian compact group. If G is any abelian compact group, then the abelian A5.0.15 Definition. group (without topology) Hom(G, T) of all continuous homomorphisms of G into T is called the dual group of the abelian compact group G and is written G. So if G is an abelian compact group, then ... |
L. Zippin and A. Gleason in the 1950s characterized noncompact Lie groups by conditions (ii) and (iii) above. Earlier we reduced the study of the topology of compact groups to the study of the topology of connected compact groups. 510 APPENDIX 5: TOPOLOGICAL GROUPS: A GRADUATE COURSE Next we reduce the study to that of... |
we shall prove this soon. Exercises A5.1 1. Let (G, τ ) be a topological group, e its identity element, and k any element of G. If U is any neighbourhood of e, show that there exists an open neighbourhood V of e such that 517 (i) V = V −1, (ii) V 2 ⊆ U , and (iii) kV k−1 ⊆ U . (In fact, with more effort you can show tha... |
uotient maps of topological groups are not necessarily closed maps. For if R2 denote the product group R × R with the usual topology, example, and p is the projection of R2 onto its first factor R, then the set S = is closed in R2 and p is a quotient map with p(S) not x, 1 x closed in R. : x ∈ R, x = 0 If G is a topolog... |
hism φ : G → T such that φ(g) = e. Case (i). Assume gn = e, and gk = e for 0 < k < n. Let H = {gm : m ∈ Z}. Define φ : H → T by φ(g) = an nth root of unity = r, say, (r = e), and φ(gm) = rm, for each m. Now extend φ to G by Proposition A5.3.6. Case (ii). Assume gn = e, for all n > 0. Define φ(g) = z, for any z = e in T. ... |
elian groups which will in future be written additively. However, we shall still refer to the product of two groups A and B (and denote it by A × B) rather than the sum of the two groups. We shall also use An to denote the product of n copies of A and Ai for the product of the groups Ai, i ∈ I. i∈I The identity of an a... |
e abelian group with finite basis. In other words, any subgroup of a free It can be shown that any subgroup of a free abelian group is a free abelian group. For details see A.G. Kurosh [250]. (iv) Finally, we record that if the abelian group G admits a homomorphism φ onto a free abelian group F then G is isomorphic to F... |
) U ∈ U =⇒ U −1 ∈ U ; (c) if U ∈ U then there is a V ∈ U such that V 2 ⊆ U ; (d) if U ∈ U and V ∈ U , then U ∩ V ∈ U ; (e) if U ∈ U and U ⊆ V ⊆ X2, then V ∈ U . The pair (X, U) is called a uniform space and each member of U is called an entourage. If R is the set of real numbers, then the usual uniformity A5.6.2 Exampl... |
dual group of R is algebraically isomorphic to R itself, under the isomorphism d → γd. To prove the claim, let K denote the kernel of γ. If K = R then Proposition A5.5.2 says that K is isomorphic to Z. Further by Corollary A5.5.5 the quotient group R/K is topologically isomorphic to T. As in (i) above there are only t... |
and γ2 ∈ Γ2 determine a character γ ∈ Γ by the formula (g, γ) = (g1, γ1) + (g2, γ2) (1) Since every γ ∈ Γ is completely determine by its action on the subgroups G1 and G2, equation (1) shows that Γ is algebraically the direct sum of Γ1 and Γ2. To see that Γ has the product topology Γ1 × Γ2 simply note that (a) P (K, Vε... |
⊆ Vn−1, for n 2 and g ∈ K, where K is a compact set which generates G. Put H = Vn and use Exercises A5.6 #3.] ∞ n=1 3. Using Exercise 2 above, deduce statement B from statement A. (A) Every compact metrizable abelian group has enough characters to separate points. 573 (B) Every compact Hausdorff abelian group has enough... |
and there are only a finite number of distinct bi. G(n) and, using the [Hint: Let G(n) = {x ∈ G : nx = 0}. Observe that G = Baire Category Theorem A5.4.1, show that one of the quotient groups G/G(n) is finite. ∞ n=1 Deduce that the orders of all elements of G are bounded. Then use the structure theorem of abelian groups... |
morphic to a subgroup of Td, the circle group endowed with the discrete topology. [Hint: Use (i) with A = Z and B = G.] (iii) Let A be an LCA-group which satisfies the duality theorem and B an LCAgroup. If f is a continuous one-one homomorphism of A into B show that f ∗(B∗) is dense in A∗. [Hint: See the proof of Coroll... |
−→ K −−→ G −−→ G/K −−→ 0 f1 f2 Applying Proposition A5.13.2 to this sequence and Proposition A5.13.3 to the dual sequence, we obtain that the sequence 0 −−→ K∗∗ −−→ Γ∗ −−→ (G/K)∗∗ −−→ 0 f ∗∗ 1 f ∗∗ 2 is also exact. It is easily verified that the diagram 0 −−→ K 0 −−→ K∗∗ αK f1 −−→ G −−→ G/K f2 −−→ 0 α ... |
pology. Saak Gabriyeylan (Gabriyelyan [146]) proved that every infinitely-generated abelian group admits a m.a.p. topology. Wigner 601 9. Let Γ be any LCA-group and γ1, . . . , γn ∈ Γ. If φ is any homomorphism of Γ into T, show that there is a continuous homomorphism ψ of Γ into T such that |ψ(γi) − φ(γi)| < ε, i = 1, .... |
s A5.12 #4, there exists a family {Yn}, n = 1, 2, . . . of compact neighbourhoods of 0 in Γ such that every compact subset of Γ lies in some Yn and Yn ⊆ Yn+1, n 1. So the family {g ∈ G : (g, γ) ∈ V1/k, for all γ ∈ Yn}, for k = 2, 3, . . . and n = 1, 2, . . . is a base of neighbourhoods of 0 in G. (Observe that in sayin... |
s G1, G2, . . . , Gn are somewhat divisible, then the product group G1 × G2 · · · × Gn is somewhat divisible. (ii) Any quotient group of a somewhat divisible group is somewhat divisible. (iii) An abelian group is divisible if and only if it is somewhat divisible. 615 A5.99 Credit for Images 1. Leon Battista Alberti. Cr... |
statements.] 622 APPENDIX 6: FILTERS AND NETS A6.1.4 Proposition. (i) If F is a fliter on a set X, then F has the finite intersection property; that is, if F1, F2, . . . , Fn ∈ F, n ∈ N, then F1 ∩ F2 ∩ · · · ∩ Fn = Ø; (ii) Let S be a set of subsets of a non-empty set X. There exists a filter F on X such that S ⊆ F if and... |
Ø. Assume that (X, τ ) is compact and let F be any filter on (X, τ ). Then F has the finite intersection property. Put G = {F : F ∈ F}. Then G has the finite intersection property too. As (X, τ ) is compact, there exists a point x0 ∈ X, such Fi∈F Fi. So if Nx0 ∈ Nx0, the neighbourhood filter in (X, τ ) of x0, that x0 ∈ the... |
ghbourhood filter at x in (X, τ ). Fxi∈Sx Proof. Exercise. Let X be a non-empty set and τ 1 a topology on the A6.1.31 Corollary. set X. Further let Sx be the set of all ultrafilters which converge to x on (X, τ 1). Let τ be the topology defined from Sx, x ∈ X, as in Proposition A6.1.30. Then τ = τ 1. Proof. Exercise. A6.1... |
1. Since F1 ∈ F1, F1 ⊆ S, and thus N ∩ S = Ø. Hence a is a limit point of S, as required. 643 Let (X, τ ) be a topological space and S a subset of A6.2.12 Corollary. X. The following two properties are equivalent: (i) S is a closed set in (X, τ ); (ii) Let F be a filterbase on (X, τ ) such that F ∈ F implies F ⊆ S. Furt... |
rected set. A6.3.4 Definition. set. Then a function φ : D → X is said to be a net in the space (X, τ ). Let (X, τ ) be a topological space and (D, ) a directed It is often convenient to write the net φ above as {xα} where φ(α) = xα ∈ X, α ∈ D. 38There are two 14 minute YouTube videos which provide an excellent introduct... |
3, there exist an α4 with α3 α4 such that xα4 ∈ U3. So (α4, U3) ∈ D2. As α1 α4, α2 α4, U1 ⊇ U3 and U2 ⊇ U3, we see that (α1, U1) (α4, U3) and (α2, U2) (α4, U3). So D2 is indeed a directed set. Define a map θ : D2 → D1 by θ(α, U ) = α, where α ∈ D1 and U is a neighbourhood of a. Clearly θ is non-decreasing. Now consider ... |
hat B = {O∗ : O ∈ τ } is a basis for a topology τ ω on ωX and that this topological space (ωX, τ ω) is a compact T1-space which has (X, τ ) as a subspace. 664 APPENDIX 6: FILTERS AND NETS We claim that for any O ∈ τ O∗ = ωX \ (X \ O)∗ . Proof of (6). ωX \ (X \ O)∗ = (X ∪ F) \ ((X \ O) ∪ {U ∈ F : X \ O ∈ U}), by (3) = (... |
computer programming for knot tabulation. World Scientific Publishers, Singapore; River Edge, N.J., 1999. [14] S.A. Argyros, P. Dodos, and V. Kanellopoulos. Unconditional families in banach spaces. Math. Ann., 341:15–38, 2008. [15] Alexander Arhangel’skii and Mikhail Tkachenko. Topological Groups and Related Structures.... |
Math. Monthly, 118:3–21, 2011. [94] Jane Cronin. Fixed points and topological degree in nonlinear analysis. American Mathematical Society, Providence, R.I., 1964. [95] G. Cybenko. Approximation by superpositions of a sigmoidal function. Math. Control Signals Systems, 2:303–314, 1989. [96] R.J. Daverman and R.B. Sher, ... |
athitrust.org/cgi/pt?id=uc1.$b417528;view=1up;seq=52. [171] Yasunao Hattori. Dimension and superposition of bounded continuous functions on locally compact separable metric spaces. Topology and its Applic., 54:123–132, 1993. [172] Felix Hausdorff. Dimension und außeres maß. Math. Annalen, 79:157–159, 1919. [173] Felix H... |
elsea Publishing, New York, 1956; Reprinted 1960. [251] N.P. Landsman. Lecture notes on C∗algebras and quantum mechanics, 1998 http://tinyurl.com/zxsg54u. [252] Serge Lang. Real and functional analysis, third edition. Springer-Verlag, Berlin, Heidelberg, 1993 http://tinyurl.com/zmxpfmv. [253] H.A. Lauwerier. Fractals: ... |
neral topology. Academic Press, New york, 1980. [337] G.M. Reed, A.W. Roscoe, and R.F. Wachter. Topology and category theory in computer science. Oxford University Press, Oxford, England, 1991. [338] Dieter Remus. Topological groups with no non-trivial characters. In: General Topology and its relations to modern analys... |
ses applications. Hermann & Cie., Paris, 1951. [422] Hermann Weyl and F. Peter. Die vollständigkeit der primitiven darstellungen einer geschlossenen kontinuierlichen gruppe. Math. Ann., 97:737–755, 1927. [423] Stuart G. Whittington, De Witt Sumners, and Timothy Lodge, editors. Topology and geometry in polymer science.... |
ions almost-periodic, 599 Fundamenta Mathematica, 433 Fundamental Theorem of Algebra, 217 GL(n, C), 513 Gδ-set, 55, 170 ΓX, 347 G-base, 313 Galileo Galilei, 395 Gauss, Carl Friedrich, 485 Gelfand-Naimark Representation Theorem, 388 general linear group, 494, 513 Generalized Heine-Borel Theorem, 184, 204 geometry descri... |
y if, 56 mathematical, 25 proper subset, 37 property separation, 49 fixed point, 120 topological, 109 protorus, 503 Q, 54, 93, 265 rationally dependent, 538 real projective plane, 350 real projective space, 350 real trignometric polynomial, 384 reduced cone, 347 reduced suspension, 347 refinement, 306, 641 reflection, 492... |
ss Approximation Theorem, 360 Weierstrass Intermediate Value Theorem, 119 upper bound, 85, 268, 269, 271, 272, Weierstrass, Karl, 354 379 least, 379 weight, 246 network, 246 upper semicontinuous, 172 well-ordered set, 269, 415 Urysohn’s Lemma, 278, 301 well-ordering, 415 Urysohn’s Metrization Theorem, 284 Well-Ordering... |
y – A connected graph Ξ. – For each vertex v ∈ V (Ξ), a path-connected space Xv. – For each edge e ∈ E(Ξ), a path-connected space Xe. – For each edge e ∈ E(Ξ) attached to v± ∈ V (Ξ), we have π1-injective maps ∂± e : Xe → Xv± . The realization of X is |X | = X = v∈V (Ξ) Xv (∀e ∈ E(Ξ), ∀x ∈ Xe, (x, ±1) ∼ ∂± e∈E(Ξ)(Xe × [... |
geodesic triangle in X is δ-slim. In this case, we say it is δ-hyperbolic. Example. R2 is not Gromov hyperbolic. Example. If X is a tree, then X is 0-hyperbolic! Indeed, each triangle looks like We call this a tripod . Unfortunately, none of these examples really justify why we call these things hyperbolic. Let’s look ... |
edded interval, which is necessarily a global notion. We want an analogous local version. However, if we want to work up to quasi-isomorphism, then we cannot go completely local, because locally, you are allowed to be anything. Definition (k-local geodesic). Let X be a geodesic metric space, and k > 0. A path c : [a, b]... |
hyperbolic spaces, this is quasi-isometry invariant. Example. If Γ = π1Σ, with Σ closed hyperbolic surface, then ∂∞Γ = S1 and the union X ∪ ∂∞X gives us the closed unit disc. Theorem (Casson–Jungreis, Gabai). If Γ is hyperbolic and ∂∞Γ ∼= S1, then Γ is virtually π1Σ for some closed hyperbolic Σ. Example. If Γ is free,... |
metric. Recall we previously stated the Hopf–Rinow theorem in the context of differential geometry. In fact, it is actually just a statement about length spaces. Theorem (Hopf–Rinow theorem). If a length space X is complete and locally compact, then X is proper and geodesic. This is another application of the Arzel´a–As... |
⊆ ˜X, there is a finite-sheeted cover X → X such that the natural covering map p : ˜X → X is injective on K. A good (though not technically correct) way to think about this is follows: if we have a map f : K → X that may be complicated, and in particular is not injective, then we might hope that there is some “finite re... |
erical space, 25 Bass–Serre tree, 27 Britton’s lemma, 28 Cartan–Hadamard theorem, 40, 45 CAT(κ) space, 41 CAT(0) group, 43 Cayley graph, 3 cocompact action, 7 comparison point, 41 comparison triangle, 41 cube complex, 50 special, 54 curvature, 41 cyclicly reduced word, 34 Dehn function, 14 Dehn presentation, 37 direct ... |
an view K as the Q-points of an n-dimensional affine group scheme. We can then define RK/QGm ⊆ A to be the set on which X is invertible, and then an algebraic Hecke character is a homomorphism of algebraic groups (TK)/C → Gm/C, where TK = ResK/Q(Gm). If we have a real place v of K, then this corresponds to a real embeddin... |
ough NK/F , where F = EH ∩ K = K ∩ QCM. Recall that a homomorphism ϕ : K × → C× is algebraic iff it is a character of the commutative algebraic group TK = RK/QGm, so that TK(Q) = K ×, i.e. there is an algebraic character ϕ : TK/C → Gm/C such that ϕ restricted to TK(Q) is ϕ. Then ϕ is of Serre type iff ϕ is a character of... |
= v∞ det(1 − q−s v Frobv|V Iv )−1 Λ(ρ, s) = LL∞ L∞ = v|∞ L(ρv, s). This is well-defined as the decomposition groups ¯v | v are conjugate. If dim V = 1, then ρ = χ ◦ Art−1 K for a finite-order Hecke character χ, and then The facts we had for local factors extend to global statements L(ρ, s) = L(χ, s). Proposition. (i) L(ρ... |
(Tr − 1)m m converges -locally, and then Tr = exp Nr. We claim that Nr is nilpotent. To see this, if we enlarge E, we may assume that all the eigenvalues of Nr are in E. For δ ∈ WF and γ ∈ IF , we know So for all γ ∈ I . So t(δγδ−1) = ω(δ)t(γ). ρ(δγδ−1) = ρ(γ)w(σ) ρ(σ)Nrρ(δ−1) = ω(δ)Nr. Choose δ lifting ϕq, w(δ) = q. ... |
is in terms of the full Weil group, we can talk about the one-dimensional representations of WF , and write local class field theory as a correspondence characters of GL1(F ) ←→ 1-dimensional representations of WF The Langlands correspondence aims to understand the representations of GLn(F ), and it turns out this corre... |
e the following definition: Definition (Square integrable representation). Let (π, V ) be an irreducible smooth representation of G. We say it is square integrable if ωπ is unitary and for all (v, ). |πv,| ∈ L2(G/Z) Note that the fact that ωπ is unitary implies |πv,| is indeed a function on L2(G/Z). In general, it is unl... |
me g ∈ WF \ IF , σ(g) has an eigenvalue of absolute value 1. 56 4 The Langlands correspondence IV Topics in Number Theory (ii) im σ is relatively compact, i.e. has compact closure, i.e. is bounded. (iii) σ is unitary. Proof. The only non-trivial part is (i) ⇒ (ii). We know im σ = σ(Φ), σ(IF ) = H, where Φ is some lift ... |
ne representation, 42 Weil–Langlands group, 46 wild inertia group, 5 63 |
D n n m! C m ; 1! m 0 n; and use this to show by induction on n that n m! D nŠ mŠ.n ; 0 m n: m/Š (b) Show that n 0 m X D 1/m . n m! D 0 and n 0 m X D n m! D 2n: (c) Show that y/n .x C D (This is the binomial theorem.) n 0 m X D n m! m: xmyn 20. Use induction to find an nth antiderivative of log x, the natural logarithm ... |
her point of S . 2 (d) x0 is exterior to S if x0 is in the interior of S c . The collection of such points is the exterior of S . Example 1.3.7 Let S ; . 1 D 1 [ .1; 2/ 3 g [ f . Then 24 Chapter 1 TheRealNumbers (a) The set of limit points of S is . ; 1 (b) @S ; . 1; 1; 2; 3 1 (c) 3 is the only isolated point of S . (d... |
R such that if H is any open covering of S , then S H comprised of finitely many open sets from H . Show that has an open covering S is compact. e 22. A set S is. in a set T if S T S . (a) Prove: If S and T are sets of real numbers and S T , then S is dense in T if and only if every neighborhood of each point in T conta... |
ter 2 DifferentialCalculusofFunctionsofOneVariable Proof From (2.1.9) and Definition 2.1.2, if > 0, there is a ı1 > 0 such that f .x/ < ı1, and a ı2 > 0 such that j < L1 j if 0 < x j x0 j g.x/ j L2 j < if 0 < x j x0 j < ı2. Suppose that so that (2.1.15) and (2.1.16) both hold. Then 0 < x j x0 j < ı D min.ı1; ı2/; (2.1.1... |
and x1 < x2: (2.1.20) In either case, f is on I . If can be replaced by > in (2.1.20), f is decreasing on I . In either of these two cases, f is strictly monotonic on I . can be replaced by < in (2.1.19), f is increasing on I . If Example 2.1.16 The function f .x/ x; 0 x < 1; 1 x 2; D ( 2; is nondecreasing on I D Œ0; ... |
minate, then ! L1, limx D ! x0 g.x/ 0, and L1=L2 is not L2 ¤ : lim x0 x ! f g .x/ D L1 L2 D ¤ (b) Show that it is necessary to assume that L2 sin x, g.x/ cos x, and x0 =2. D D 0 in(a) by considering f .x/ D 31. Find (a) lim 0 x C ! x3 2x4 C C (c) lim x !1 2x4 x3 C C 3x2 2x 2x 3x2 3 2 C C 2 C 3 C (e) limx .ex2 ex/ !1 32... |
defined on T by Df whenever x 2 2 ı g/.x/ .f ı D f .g.x//: f .x/ D log x and g.x/ Example 2.2.8 If then Since g.x/ > 0 if x 2 .0; / 1 Df T D . D and Dg D 1; 1/, the composite function f ˚ 1 ˇ ˇ x2 : ı f is defined on .0; 1=e/ g/.x/ log D 1 f /.x/ 1 D 1 .log x/2 : .f We leave it to you to verify that g .g ı ı Section 2.2... |
C x j t j < 2ıt and x Œa; b: 2 (2.2.9) H It t 2 D Œa; b ˇ is an open covering of Œa; b. Since Œa; b is compact, the Heine–Borel theorem implies that ˇ there are finitely many points t1, t2, . . . , tn in Œa; b such that It1, It2 , . . . , Itn cover Œa; b. Now define ˚ We will show that if ı min f ıt1; ıt2; : : : ; ıtn g... |
28. Let f and g be uniformly continuous on an interval S . (a) Show that f g are uniformly continuous on S . (b) Show that fg is uniformly continuous on S if S is compact. (c) Show that f =g is uniformly continuous on S if S is compact and g has no g and f C zeros in S . (d) Give examples showing that the conclusion of... |
f .x/ x f .x0/ x0 C if the limit exists, while if f is defined on .a; x0, the left-hand derivative of f at x0 is defined to be f .x/ x f .x0/ x0 lim x0 ! if the limit exists. Theorem 2.1.6 implies that f is differentiable at x0 if and only if f 0 C and f 0 .x0/ exist and are equal, in which case f 0 .x0/ D x .x0/ .x0/ .... |
f is continuous and increasing on Œa; b. Let f be differentiable at a 0. If g is the inverse of f Theorem 2.2.15), show point x0 in .a; b/, with f 0.x0/ that g0.f .x0// (a) Show that f 0 C (b) Example 2.3.4 shows that f 0 C / exists but f 0 C (c) Complete the following statement so it becomes a theorem, and prove the ... |
t. However, ! f .x/ g.x/ D lim 0 x ! lim 0 x ! 1 x sin.1=x/ .sin x/=x D 1 1 D 1: The Indeterminate Form 0 1 We say that a product fg is of the form 0 0 and the other approaches L’Hospital’s rule after writing ˙1 as x 1 b ! b if one of the factors approaches In this case, it may be useful to apply ! as x . f .x/g.x/ f .... |
2.3.2. Lemma 2.5.2 If f .n/.x0/ exists; then f .x/ n D 0 r X D f .r /.x0/ r Š .x x0/r C En.x/.x x0/n; (2.5.7) where lim x0 x ! En.x/ En.x0/ 0: D D Proof Define En.x/ f .x/ .x Tn.x/ x0/n D 8 < 0; Section 2.5 Taylor’sTheorem 101 x0 ; g f ; x x Df x0: 2 D Then (2.5.5) implies that limx (2.5.7). x0 En.x/ : ! D En.x0/ D 0, a... |
1) at x0 if f is differentiable in a neighborhood of x0 and f .x0/ ¤ (a) Prove that f has a simple zero at x0 if and only if 0, while f 0.x0/ D 0. f .x/ D g.x/.x x0/; where g is continuous at x0 and differentiable on a deleted neighborhood of x0, and g.x0/ 0. ¤ (b) Give an example showing that g in(a) need not be diff... |
tition point of P is also a partition point of P 0; that is, if P 0 is obtained by inserting additional points between those of P . If f is defined on Œa; b, then a sum D n 1 j X D f .cj /.xj xj 1/; where xj 1 cj xj ; 1 j n; is a Riemann sum of f over the partition P . (Occasionally we will say D f more simply that is a... |
than k.=2k/ =2, because of (3.1.10). Since Mj < =2 for all other values of j , the sum of the other terms is less than =2, and Mj D 2 n .xj 1 j X D xj 1/ D 2 .xn x0/ D 2 .2 1/ D 2 : Therefore, S.P0/ < and, since can be chosen as small as we wish, no positive number is less than all upper sums. This proves (3.1.9). The ... |
inequality implies that b a Z f .x/ dx a Z b f .x/ dx b a f .x/ dx Cˇ ˇ ˇ ˇ Now suppose that > 0. From Definition 3.1.3, there is a partition P0 of Œa; b such that ˇ Z a S.P / ˇ ˇ ˇ ˇ : f .x/ dx ˇ ˇ ˇ ˇ ˇ ˇ C j S.P / j (3.2.7) b f .x/ dx b S.P0/ < f .x/ dx a 3 : C Z From Definition 3.1.1, there is a ı > 0 such that a Z b... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.