Let (X, d) be a metric space. Suppose that (xn) is a sequence of points in X which converges to a point x E X, and (yn) is a sequence of points in X which converges to a point y E X. Prove that the sequence of real numbers, d(xn, Yn) coverges to d(x, y).

Q: Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 -…

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Q: Let X be a normed space and let 0x0 EX. Show that there is f = X* such that f(x) = 1 and ||f|| =…

A: Given, a normed space  and .It is required to show that, there exists  (the dual space of ) such…

Q: Suppose f and g are continuous complex functions on a metric space X. Prove f + g is continuous on X

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Q: 2. Let f and g be continous mappings on a metric space X into a metric space Y and E be a dense…

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Q: Let (X,d) be a metric space. For x,y ∈ X, let D(x, y) = √d(x, y). Prove that D is a metric on X.…

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Q: (a) Determine whether or not it is convex or concave. A = {(x, y) = R²|yx4, and -2≤ x ≤ 2} B = {(x,…

A: Necessary Sufficient Nature f'x0=0 f'x0=f''x0=f'''x0=..=fn-1x0=0  fnx0>0 n even Convex f'x0=0…

Q: (d) Show that T is bounded and find ||T||. (e) Find the adjoint T* of T.

A: We will use the vector/matrix notations to visualise things easily.

Q: (a) In a metric space (M,d), prove that the sequence (b) = (b, b,b,...) is converge to the point b.…

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Q: Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f : (a,…

A: Given:f:a, b→ℝ is an unbounded function. To show : f' is also unbounded On the contrary,  assume…

Q: Given points p = and p3 = in R?, 2 P2 = let S = conv {p],P2» P3}. For each linear functional f, find…

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Q: Let (X, d) be a metric space. Suppose that (xn) is a sequence of points in X which converges to a…

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Q: Let (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as 8(M)= sup d(x,…

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Q: Define a metric as d(x,y)=|x1-y1|+|x2-y2|. Find the set of all points x=(x1,x2) in R2 such that…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.