Problem 5. Let (X, d) be a metric space, and E C X. Show that E =N{B|ECB and B is closed}. Remark: You can show this in essentially two different ways: using sequences, or the original definitions with open sets. It is a good exercise for yourself to try both!

Q: Real Analysis We started metric spaces yesterday.  I am asked to prove that the metric space defined…

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Q: Find real numbers x and y such that iy 2 - i i + 3 1 + i ||

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Q: 2. Give an example in which T is injective but T is not surjective. (Take, e.g., X = L²([0, 1]) and…

A: As per the question we have to find the solution to the part 2 which asks to give an example in…

Q: If 7'(t) = (cos t, e²“, 3t²) and 7(0) = (3, 1, –2), find the function 7'(t).

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Q: find the length of the side of triangle ABC and determine the type of the resulting triangle.…

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Q: Using only the definition of RS-integral, evaluate .2

A: See below

Q: Let X be the space of reals with the topology T = {X, [0, 1], 0) (Example 2.1(b)), and let A = [-1,…

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Q: 11. Let (X, d) be a pseudometric space, and suppose d has the property that d(a, b) >0 whenever ab.…

A: 11) To prove that every finite subset of X is closed, we need to show that its complement is open.…

Q: Theoremi I f f is a ConveX fPanchions Then the level set Cx is a Convex set

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Q: 3. Determine which of the following sets of functions are subspaces of F[a, b]. a) All functions f…

A: NOTE: According to guideline answer of first three subpart can be given, for other please ask in a…

Q: 1. (a) Prove that if T(X) is dense in Y, then the adjoint T : Y* → X* is injective. (b) Prove that…

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Q: 3. Suppose that p is a nonzero continuous operator such that p is Hermitian (i.e. p* = p). (a) Show…

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Q: 2) P(V,T) = "RT (assume the number of particles/moles n is constant)

A: write down the total differential of the following function P(V,T) = nRTV.

Q: 5) Find the unit vector in the direction of v = 2i + j+k 1 - i+-j+ d) i+j+k 1 1 j+ V5 а) V5 k b)

A: We can solve this as follows:

Q: Use Cramer's Rule to solve the system. X₁ - 2x₂ = 1 4x₁ + 5x₂ = -5 (X₁₁ X₂) = ( (x1, 15 1 13' 13 X

A: We know thatCramer's rule :For the  following system

Q: Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | x² + y² = 1}. Let U₁ = {(r,y) € S¹…

A: As you have asked, we construct the functions which give atlas of the smooth manifold.

Q: Let f be the map Sanding each complex pumber Show that fl.z)-f2)f(2)

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Q: Use the divergence theorem to find the flux of F = [x²yz, -xy² z, 6 – 5z2| on the hemisphere S =…

A: Consider the given function. The divergence theorem is defined as. Now, find the value of div F.

Q: Problem 1. Let H be a Hilbert space and T: H → H be a linear application. Suppose that (Tx, x) > 0…

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Q: 2. Deduce that there exists a positive integer N such that f'oN for all feX such that flm 1.

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Q: Let X be the set of real numbers, and let d be the pseudometric given by d(z. y) = 1 if z #y, and…

A: As per the question we are given the set X of real numbers and the pseudometric d defined on X by…

Q: tf 169

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Q: 4. Let 0 <r <1 and let T be the smooth manifold of R³ defined by the equation: (z² + y² +r²-2²-1)² -…

A: note :As per the instructions we will only solve parts (4), (5), (6) of the question.

Q: VECTOR SPACE Which one is the closed vector? a.V{1} b.V {0}

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Q: §8.3, Exercise 4. Let E¡j be the 2 × 3 matrix whose entry in the ith row and jth column is 1 and all…

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Q: Topology Let X and Y be two compact spaces, show that X×Y is compact.

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Q: 1. Suppose f: [a, b].→ R is C¹ and that f(a) = f(b) = 0. Show that 2 (S f(x)dx) a 6³ - a³ 3 rb […

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Q: Limits

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Q: 1. (a) Prove that if T(X) is dense in Y, then the adjoint T* : Y* → X* is in (b) Prove that if T* is…

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Q: Problem 1. Let H be a Hilbert space and p: HH be a projection, i.e. it is a linear application such…

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Q: Wueslien no 0BN Banach space 3. Show that R is a

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Q: Show that sin x dx=1-cos t (0<x<t<}r).

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Q: Question M = (M,d) and Y = (Y,d~) denote metric spaces. Let f: M→ Y be a Lipschitz map. Let a,e E Y,…

A: Given: and denote a metric space, let  be a Lipschitz map. Let . Let . Let . Assuming that and To…

Q: Are the functions ³ y₁(t) = 1−³, y₂(t) = = t a fundamental set of solutions of t²y + 2ty -6y=0

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Q: Theorem of let f: (X₁, T₁) (X₁ste) be a homeomorphism and ACX₁ Then (A₁, ZA₁) = (F(A)), Z (A))

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Q: (b) Show that, for all operator S: H→ H, we have Re[(Sx, y) + (x, Sy)]| ≤ 2v(S)||1|||1||. (e) Deduce…

A: Given Information:Let  be a hilbert space. For all operator T on H, we define .Suppose that is a…

Q: If T is a mixed tensor, then T} + T{ is generally not a tensor a tensor May or may not

A: Answer : may or may not  Hence, Option (iii) is true.

Q: Problem 1. Let (X, dx), (Y, dy) be metric spaces, and equip R with its usual absolute value-metric.…

A: Given: Let us consider the given,X,dX,Y,dY a) Let us determine the sequence of functions fn:ℝ→ℝ b)…