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![Now take 2=1 (1-d, 1+8) nA = (-4,1) = infinite 1 E A Now take & E (R-[0, 1] (2-8,2+8) n A = empty & infinite && A 22613 = Hen](http://img.homeworklib.com/questions/50036b30-0e7d-11ec-8d34-018aa795710a.png?x-oss-process=image/resize,w_560)
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4. Let A = [0,1) CR, where R is endowed with its usual metric. (a) What...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Def...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
- Let V be the vector space of continuous functions defined f : [0,1] → R...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
Let X be a metric space and let E C X. The boundary aE of E...
Let X be a metric space and let E C X. The boundary aE of E is defined by E EnE (a) Prove that DE = E\ E°. Here Eo is the set of all interior points of E; E° is called the interior of E (b) Prove that E is open if and only if EnaE Ø. (c) Prove that E is closed if and only if aE C E (d) For X R find Q (e) For X...
1. (a) Let d be a metric on a non-empty set X. Prove that each of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
Let A be a subset of R. If x € R we say x is a...
Let A be a subset of R. If x € R we say x is a boundary point of A if for all € > 0, (0 – €,x +E) NA # and (x - €, x+) NĀ+). The boundary of A is a A. It is the set of all boundary points of A. The interior of A is int A = A (BA). The closure of A is cl A = AU (DA). Let B be a subset...
REAL ANALYSIS Question 1 (1.1) Let A be a subset of R which is bounded above....
REAL ANALYSIS Question 1 (1.1) Let A be a subset of R which is bounded above. Show that Sup A E A. (1.2) Let S be a subset of a metric space X. Prove that a subset T of S is closed in S if and only if T = SA K for some K which is closed in K. (1.3) Let A and B be two subsets of a metric space X. Recall that A°, the interior of A,...
4. Let S = {1,2,3). Define a relation R on SxS by (a, b)R(c,d) iff a...
4. Let S = {1,2,3). Define a relation R on SxS by (a, b)R(c,d) iff a <c and b <d, where is the usual less or equal to on the integers. a. Prove that R is a partial order. Is R a linear order? b. Draw the poset diagram of R.
(7) Let R= {f [0,1] - R | f continuous} be the ring of all continuous...
(7) Let R= {f [0,1] - R | f continuous} be the ring of all continuous functions from the interval [0,1] to the real numbers. (a) For cE [0, 1, prove that Me := {feR | f(c) = 0} is a maximal ideal of R. Hint: consider the evaluation map ec- (b) Show that if M is any maximal ideal of R then there exists a cE [0,1 such that M = Me. Hint: show that any maximal ideal M...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
Q7 Let A, B C M where M is metric space. Suppose there exist open sets , V C M such that A C B C V and V-0. Prove that A and B are separated. Q7 Let A, B C M where M is metric space. Suppose the...
Q7 Let A, B C M where M is metric space. Suppose there exist open sets , V C M such that A C B C V and V-0. Prove that A and B are separated.
Q7 Let A, B C M where M is metric space. Suppose there exist open sets , V C M such that A C B C V and V-0. Prove that A and B are separated.
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
Let X be a metric space and let E C X. The boundary aE of E is defined by E EnE (a) Prove that DE = E\ E°. Here Eo is the set of all interior points of E; E° is called the interior of E (b) Prove that E is open if and only if EnaE Ø. (c) Prove that E is closed if and only if aE C E (d) For X R find Q (e) For X...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
Let A be a subset of R. If x € R we say x is a boundary point of A if for all € > 0, (0 – €,x +E) NA # and (x - €, x+) NĀ+). The boundary of A is a A. It is the set of all boundary points of A. The interior of A is int A = A (BA). The closure of A is cl A = AU (DA). Let B be a subset...
REAL ANALYSIS Question 1 (1.1) Let A be a subset of R which is bounded above. Show that Sup A E A. (1.2) Let S be a subset of a metric space X. Prove that a subset T of S is closed in S if and only if T = SA K for some K which is closed in K. (1.3) Let A and B be two subsets of a metric space X. Recall that A°, the interior of A,...
4. Let S = {1,2,3). Define a relation R on SxS by (a, b)R(c,d) iff a <c and b <d, where is the usual less or equal to on the integers. a. Prove that R is a partial order. Is R a linear order? b. Draw the poset diagram of R.
(7) Let R= {f [0,1] - R | f continuous} be the ring of all continuous functions from the interval [0,1] to the real numbers. (a) For cE [0, 1, prove that Me := {feR | f(c) = 0} is a maximal ideal of R. Hint: consider the evaluation map ec- (b) Show that if M is any maximal ideal of R then there exists a cE [0,1 such that M = Me. Hint: show that any maximal ideal M...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
Q7 Let A, B C M where M is metric space. Suppose there exist open sets , V C M such that A C B C V and V-0. Prove that A and B are separated.
Q7 Let A, B C M where M is metric space. Suppose there exist open sets , V C M such that A C B C V and V-0. Prove that A and B are separated.
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