Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x, y) = supr, - Ynl- Show that (B, d) is a metric space.

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A: A function is said to be bijective if the function is both one one and onto. The given function is…

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A: Please see the attachment

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A: Given that X is a normed linear space over F and X* = BX , F is a Banach space. ( X* is called the…

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Q: Let X be a locally compact Hausdorf f space

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A: Here, in the question, there are given the bounded linear operators l1→l∞ which are given as…

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Q: 2. Let f : [a, b] R be a bounded function. Prove that if f is continuous at ro € [a, b), then L(|f])…

A: Here we use the property of continuty.

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A: given let X be a set , let fn:X→ℝ be a sequence of function and let f:X→Y be a function. let ρ¯ be…

Q: (b) Give an example of a bounded function f on [a,b] for which f2 e R[a,b), but f is not in R[a,b).

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Q: (3.2) Let f :X Y be a continuous function between metric spaces and let {rn} be a sequence of points…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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Q: prove that any Continuouse function [a,b] en is measurable.

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Q: Let f: X_nte, Y be X,Y are Tj spaces. show that: Xö locally compact iff Y is so . perfect function,…

A: Given: f:X→ontoY is a perfect function, where X,Y are T2-spaces. To show that X is locally compact…

Q: Let f be a bounded function on [a, b]. If P and Q are partitions of [a, b], prove that L( f , P) ≤…

A: Given f be a bounded function on a,b. If P and Q are the partitions of a,b. Then We have to prove…

Q: a) Let fn: [a, b] → R be a sequence of integrable functions, converging uniformly to f: [a, b] → R.…

A: As per our guidelines we are supposed to answer only one question. For the solution of the first…

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A: Given that, P denote the set of function f in C0,1

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Q: Define f : R R by if a 20 = |x| = (x)S -a if r < 0. Use the sequence characterization of continuity…

A: Given function is  Now, by the definition sequence characterization of continuity  If  be a…

Q: Let f : [a, b] → R be a continuous function and suppose that N is a natural number. If ¤1,..., IN…

A: It's a question from intermediate value theorem, so let's see how we can do it.

Q: (5) Prove that Lp(f)<Uq(f) for any bounded function f and partitions P,Q of [a, b]. (Hint: Consider…

A: Note: According to Bartleby guidelines; for more than one question asked only the first one is to be…

Q: Let V be the F-vector space of F-valued sequences

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A: Given that, P denotes the set of functions f in C0,1

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Q: Suppose that fn ⇒ f on R and that each fn is a bounded function. Prove that f is bounded.

A: Solution

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A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

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Q: 2. (a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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A: Yes, it is always true. See the proof.

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A: Intermediate value theorem : Suppose be a continuous function then it takes all values between .

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Q: Assume that is a continuous real function on (a, b) such that 1 1 o ( x + ³) ≤ ²}/20 (x) + = 0 ( for…

A: Here given that φ is a continuous real function on a, b such that φx+y2≤12φx+12φx

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A: Given that X,d be a metric space. Suppose that xn is a sequence of points in X which converges to a…

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A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

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Q: Prove that if f is a continuous function with domain 0, 1] and X is the set of all x in 0, 1] such…

A: Given that f is a continuous function with domain 0,1. Let, X=x| x∈0,1 and fx=0. Also, given that s…

Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.