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3. (Contractible spaces.) (i) Recall that a space X is said to be contractible if it is homotopy equivalent to a point. Prove the following (a) A space X is contractible if and only if it the identit...
Topology 3. Either prove or disprove each of the following statements: (a) If d and p...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
1- Prove or disprove. (X,Y are topological spaces, A, B are subsets of a topological space...
1- Prove or disprove. (X,Y are topological spaces, A, B are subsets of a topological space X, Ā denotes the closure of the set A, A' denotes the set of limit points of the set A, A° denotes the interior of the set A, A denotes the boundary of the set A.) (a) (AUB) = A'U Bº. (b) f-1(C') = (F-1(C))' for any continuous function f :X + Y and for all C CY. (c) If A° ), then A°=Ā.
2. [2+9+6=17] Let X be a nonempty set. Two metrics d and d on X are...
2. [2+9+6=17] Let X be a nonempty set. Two metrics d and d on X are said to be uniformly equivalent if the identity map from (X, d) to (X, d) a nd its in- verse are uniformly continuous. (a) Prove that uniform equivalence is indeed an equivalence relation on the class of metrics on X. (b) Let (X,d) and (x, ) be uniformly equivalent. Are the following true or false? (i) If (X, d) is bounded, then must also...
3. (a) Prove the following: Cantor's Intersection Theorem: Let (X, d) be a complete metric space...
3. (a) Prove the following: Cantor's Intersection Theorem: Let (X, d) be a complete metric space and {Anymore a nested sequence of non-empty closed sets whose diameters D(An) have limit 0. Then An has exactly one member. csc'anno proach onsdelered. c) Show that, in part (a), n A, may be empty if the requirement that the diameters
EXERCISE 6.3.8 For each of the following spaces, which has a deformation retract of (i) a point, ...
EXERCISE 6.3.8 For each of the following spaces, which has a deformation retract of (i) a point, (ii) a circle, (iii) a figure eight, or (iv) none of these? a. R3 minus the nonnegative x, y, and z axes b. R2 minus the positive x axis c. S U(r, 0)-1 <x <1], where Sl is the unit circle in the plane d. IR3 e. S2 minus two points f. R2 minus three points g. S2 minus three points h. T...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
Problem 3. (1) Let H be a Hilbert space and S, TE B(HH). Then, prove that...
Problem 3. (1) Let H be a Hilbert space and S, TE B(HH). Then, prove that ||ST|| ||||||||| (2) Let X, Y be Hilbert spaces and Te B(X,Y). Then, prove that ||1||| sup ||T3|1 TEX=1 Let X, Y be Banach spaces. Definition (review) We denote by B(X, Y) a set of all bounded linear operators T:X + Y with D(T) = X. B(X, Y) is a vector space. Definition (review) A linear operator T:X + Y is said to be...
This was the answer I got, teacher said it was wrong Teacher said, couldnt run the...
This was the answer I got, teacher said it was wrong
Teacher said, couldnt run the gate because there wasnt any
switches
5. Design and test a simplified logic circuit to identify all numbers in the output range of function: F(x) = 2x+3 for an input domain between 0 and 6. Be sure to include your truth table. Normal 1 No Spac... Heading 1 Head Paragraph Styles t Draw Simulate View Window Help 39 ) ) 11:55 1 esu.desire2learn.com Boolean...
Please answer C 3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which...
Please answer C
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
Please answer D 3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which...
Please answer D
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
1- Prove or disprove. (X,Y are topological spaces, A, B are subsets of a topological space X, Ā denotes the closure of the set A, A' denotes the set of limit points of the set A, A° denotes the interior of the set A, A denotes the boundary of the set A.) (a) (AUB) = A'U Bº. (b) f-1(C') = (F-1(C))' for any continuous function f :X + Y and for all C CY. (c) If A° ), then A°=Ā.
2. [2+9+6=17] Let X be a nonempty set. Two metrics d and d on X are said to be uniformly equivalent if the identity map from (X, d) to (X, d) a nd its in- verse are uniformly continuous. (a) Prove that uniform equivalence is indeed an equivalence relation on the class of metrics on X. (b) Let (X,d) and (x, ) be uniformly equivalent. Are the following true or false? (i) If (X, d) is bounded, then must also...
3. (a) Prove the following: Cantor's Intersection Theorem: Let (X, d) be a complete metric space and {Anymore a nested sequence of non-empty closed sets whose diameters D(An) have limit 0. Then An has exactly one member. csc'anno proach onsdelered. c) Show that, in part (a), n A, may be empty if the requirement that the diameters
EXERCISE 6.3.8 For each of the following spaces, which has a deformation retract of (i) a point, (ii) a circle, (iii) a figure eight, or (iv) none of these? a. R3 minus the nonnegative x, y, and z axes b. R2 minus the positive x axis c. S U(r, 0)-1 <x <1], where Sl is the unit circle in the plane d. IR3 e. S2 minus two points f. R2 minus three points g. S2 minus three points h. T...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
Problem 3. (1) Let H be a Hilbert space and S, TE B(HH). Then, prove that ||ST|| ||||||||| (2) Let X, Y be Hilbert spaces and Te B(X,Y). Then, prove that ||1||| sup ||T3|1 TEX=1 Let X, Y be Banach spaces. Definition (review) We denote by B(X, Y) a set of all bounded linear operators T:X + Y with D(T) = X. B(X, Y) is a vector space. Definition (review) A linear operator T:X + Y is said to be...
This was the answer I got, teacher said it was wrong
Teacher said, couldnt run the gate because there wasnt any
switches
5. Design and test a simplified logic circuit to identify all numbers in the output range of function: F(x) = 2x+3 for an input domain between 0 and 6. Be sure to include your truth table. Normal 1 No Spac... Heading 1 Head Paragraph Styles t Draw Simulate View Window Help 39 ) ) 11:55 1 esu.desire2learn.com Boolean...
Please answer C
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
Please answer D
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
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