Let (X, T) and (Y, T₁) be two topological spaces and let f be a continuous mapping of X into Y. O None of the choices. O If f is onto and (Y,T_1) is a Hausdorff space then (X,T) is Hausdorff. If f is one-to-one and (Y,T_1 ) is a Hausdorff space then (X,T) is Hausdorff. If (Y,T_1) is a T_1 space then (X,T) is a T_1 space.

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: Q2. Let X = {a, b, c, d, m, n}, 7 = {6, X, {c}. {m,n}, {a, b, c, d}, {c, m, n}}. and Y- (b,c, m) C…

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

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Q: (a) Let X and Y be Banach spaces, T: X→ Y be a bounded linear map. Show that (i) and (ii) are…

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Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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Q: A: For each of the following maps T: V→ W, find the spaces V and W and say if they are linear or not…

A: The given map T:V→W such that Tx, y=1, y. We have to check whether the map is linear or not.

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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Q: Verify that {u₁,4₂} is an orthogonal set, and then find the orthogonal projection of y onto…

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Q: Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric…

A: A matric space X is separable if it has a countable dense subset.

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: 2. (§1.1) Sketch the following set of points in the x-y plane. {(x, y) = R²: (y - x²) (y + x²) < 0}.

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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: the quotient space of a T0−space necessarily T0

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Q: Which of the following maps from R³ to R³ are not linear? U □ f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

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Q: Let f: X → Y and g : Y → Z be functions. (a) Show that if g of is one-to-one (or "injective"), then…

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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: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

A: (a) Given a Banach space  and  the usual normed space, we want to show that the norm  is complete on…

Q: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

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Q: Let f, g,h: A→ A be any bijective functions. Then (f ogoh)-

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Q: 20. Let V be an inner product space. Prove that (a) |||x|| - ||y||| ≤ ||x − y||. (b) (x, y) =…

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Q: Determine the values of y such that D(9,3,-2), E(3,4,7) and F(21,y,-20) are collinear

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Q: (a) A¹\{1} (b) V(x+x) CA² (c) V(x2-x7,x3-x})\{0} CA³

A: note : Since you have posted question with multiple sub parts, we will provide the solution only to…

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

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Q: (b) Let X = {0, 1} and let (X*,d) be metric space, where d(x, y) = the number of elements of the set…

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Q: Let f be a continuous functio to R. Let a and b be two distinct ntained in U and fbe differentiable…

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Q: Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If…

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Q: Let A: X₁ ≤X → Y be a closed operator where X and Y are Banach spaces. Define ||x|| = ||x||+||Ax||,…

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Q: 11. Prove that the function d: Cx CR defined by d(x, y) = Solle is a metric on the set of all…

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Q: 5. In any metric space (M, D), prove that D(a1, an) ≤ D(a₁, a2) + D(a2, a3) + + D(an-1, an), for a1,…

A: Given M,D is a metric space.   The triangle inequality holds in a metric space, which means for…

Q: We note that ℤ2 which means ℤ*ℤ. Let d: ℤ2 *ℤ2→ℝ≥0 be the function that is given by…

A: The given in this question is a function d that takes two points (x0,y0) and (x1,y1) in the set of…

Q: X2 - X1' Let L:R3 → R³ be a linear operator such that L ×2 ) =| X3 - X2 , then Ker(L) spanned by: X3…

A: we know that Ker(L) is solution of the system L(X)=0  Reveal sold the system and find the ker(L)…

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

A: We have to give an example of a topological space X such that the diagonal is not closed in X×X

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: 1. Consider the following three domains in the space: {(x,y,z) E R³:1 ≤|x+y+z<4}, {(x,y,z) € R³ : 2…

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Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.