1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is continuous if O2 ⊂ O1. (2) Show that the identity map idX : (X, O1) → (X, O2) is not continuous if O1⊂≠O2.

Q: The map f:R³ f(x,y,z) = (x+y+z,0,2) O True False R³ is a linear map. defined by

A: Introduction: A mapping f: R→R is know as linear mapping if it justifies two conditions, (i)…

Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: Show that the operator U is a partial isometry U: L² ([0,1]) → L² ([0,1]), Uƒ(x) = X[0,1] ƒ(x)

A: To show that the operator U:L2([0,1])×[0,1]→L2([0,1]) defined by Uf(x)=χ[0,1](x)⋅f(x) is a partial…

Q: Let X, Y be Banach spaces and U be an open subset of X. +1)-times continuously differentiable…

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Q: Let f = 12r + kry + Jryz re-express f O f(r,y,z) = (x, ry, Tyz+) O {r,y) = (x, ry, ryz*) O f(x,y) =…

A: GIVEN f= ix+kxy+jxyz4     ....(1)

Q: The rectangular coordinates of a point are given. Plot the point. (-6/2, -6/2) y y y 10 10 10 5 5…

A: Introduction: Rectangular coordinates to polar coordinates: To convert the rectangular coordinates…

Q: or not. (c) Let f be a map from a space(X, ). into a space (Y,T) and t,T such that TSt and TS,, show…

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Q: Let L: R→ R be a linear operator, if L(X)= L(Y), then X= Y. Select one: O True O False

A: Given Linear operator is  L:R3→R3 we have to check if L(X)=L(Y) the X=Y

Q: The mapping $ (y) = fy' : H→H* defined by is not a linear m

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Q: (b) If (X ,7) is a topological space, then the inclusion function i :X →X where i (x ) =x for each x…

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Q: 4) Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

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Q: Let G : R? → R³ and F : R³ → Rª be defined by G( (x, y)) F( (a,b, c) ) (x +y, x – y, 2y) (a + b+ c,…

A: Given G:R2→R3 is defined as Gx,y=x+y,x-y,2y Also,  F:R3→R4 defined by Fa,b,c=a+b+c,2a+c,b+2c,a+b.

Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: Let ƒ: R² → R² be an isomorphism where ([-])-[²] and f([:]) = [] = Find f ¹ƒ( [7¹]) X 0 2 2 0

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Q: For the system Jx- x - Y -2x + 2y If there is a solution of the form O-1 0 1 =-2 - = 4 O 3 0 5 O…

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Q: Let f : R → R be defined by f(x) = x^2. Is f one-to-one? Explain your answer.  Is f onto? Explain…

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Q: af Ꭷf f(x, y) — f(y, x) = (x − y) — (x, y) + (y − x) — (x², y") ду or some point (x*, y*) € R² b)…

A: aSince given partial differential equation is               fx=fy⟹      fx-fy=0Auxillary equation…

Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

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Q: Suppose that R³ is given the norm ||(x, y, z) || = |x|+|y| + |2| and let S = {(x, y, z) = R³: 2x + y…

A: Given that R3 has a norm ||(x,y,z)||=|x|+|y|+|z|.The given set is S={(x,y,z)∈R3:2x+y−3z=0}.Now…

Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

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Q: b) S : M3,3 → M3,3 defined by S(A) = –A"? (i.e. negative transpose)

A: This is a question of linear transformation. The problem is to verify whether the transformation is…

Q: 3) Prove that the multiplication map · : Z/nx Z/n → Z /n given by [x] · [y] = [xy] is well-defined.

A: To prove that the multiplication map · : ℤn×ℤn→ℤn given by    x·y=xy  is well-defined.

Q: Let T: R3 -> R2 be defined by T(a, b, c) = (a + b, 2a —- c). DescribeT-1(1, 11).

A: T: R3 -> R2 Hence, T-1 : R2 ----> R3 So, necessarily we have to find a, b & c such that…

Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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Q: . Let B(X, Y) denote the space of all bounded linear operators between the normed spaces X and Y.…

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Q: Let f: R2 R2 be defined f(x, y) = (3x + y,x+y). (a) Prove that f is a bijection. (b) Keep f the same…

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Q: 2) Let X be normed space and xo + 0 be an element of X. Then using Hahn-Banach Theorem (Normed…

A: We will construct a functional satisfying given data

Q: a) Let f: R - [1, co), where f (x) = x2 + 1. Determine whether the function maps onto the given…

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Q: sup norm

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Q: f: Z→ No, x→ 2x, -2x - 1, when falls x ≥ 0 falls x < 0 مید ماس

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Q: on 1: a) Let O be the set of odd numbers and O' = {1,5, 9, 13, 17, ...} be its subset. Define the…

A: Since the part b is independent of the first part so I am answering the first question only for the…

Q: f: R →R by f(x)= x^2 and V is a subset of the codomain. Find a set V such that V has only one…

A: We are given that f : R → R by f(x) = x2 and V is a subset of the codomain. We need to find the set…

Q: 4. Show that the function : R² × R²: → R given by =u²[¹7¹] inner product on R². v does not define an

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Q: (b) Show that xE R, x2 - 4x + 5 > 0.

A: As this is a multiple question type, only first question is to be solved, according to the Bartleby…

Q: 5. Let U and V be the relations in R defined by U={(x,y):x² +2y=5} and ={(x,y):2x-y=3}. a) Find VoU.…

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Q: Consider the relation R = {(x,x): x € Z} on Z. Is R reflexive? Symmetric? Transi- tive? If a…

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Q: EHET X1 X2 - X1 Let L:R3 → R³ be a linear operator such that L( x2 ) =| x3- X3 - X2 , then Ker(L)…

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Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

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Q: Let X be a normed linear space and let P, Q E B(X). Show that the transformation T: B(X) B(X)…

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Q: Consider X = [1,2]? with the relation (x1, y1)~(x2, Y2) if x1Y2 = x2y1. Also define 3x g: X → [1,2]…

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Q: Let f be a function that takes elements of R?rR? into R' as follows: f(u, v) = |u1v1|+ |u2v2|, where…

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Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

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Q: Show that which of these relations on the set of all functions on Z-→Z are equivalence relations?…

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