Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle Inequality.]

Q: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

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Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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Q: 3. Consider the Cauchy-Schwarz inequality. That is, for u, v E R": Use some reasoning, or do some…

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Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A: 3.3 is also known as Hausdorff Property. d(x1, x2) &gt; 0 for every x1 ≠ x2   (metric space…

Q: Prove that the open ball B(a,r) is a subset of the interior of the closed ball C(a,r)

A: Given that Ba,r is an open ball of radius r and centered at a. Then a∈Ba,r                ...... (1)…

Q: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

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Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: Hw/ Find the foutier Seties of the periudic funtin %3D Ans/ 3. an-

A: According to question,Given Function,fx = x2,0&lt;x&lt;2π

Q: 4) Consider the collection of intervals on the real line: B = { (a, b) | a 1}. Is this a basis for…

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Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

Q: 3.2 Prove that in any metric space (S.) every closed ball Se[xo] is a closed set.

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Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

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Q: 3.2 Prove that in any metric space (S, d) every closed ball S,ro] is a closed set.

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Q: Prove the Bonferroni's inequality: P(EF) > P(E) + P(F) – 1

A: We know that  P(E U F) = P(E) + P(F) - P(E ∩ F)

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: Show that a set {V1, ..., Vp} in R" is affinely dependent when p > n + 2.

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Q: Prove that: m*(A) 2 0, for all sets A

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Q: Let u = and v =| [:]} Use the Cauchy-Schwarz inequality to show that a² + b? 2 + (*) a +b 2 VI

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Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: يثبت The Cauchy-Schwarz inequality states that (f, g)| ≤ ||f||||g|| for every f, g ¤ H.

A: Using the property of inner product

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Q: If a set S has both a minimum and a maximum, then S is closed.

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Q: Verify the discrete metric d2 satises the triangle inequality.

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Q: C.1 Let r > 0 and define the closed ball by B,(x) = {y € X : d(x,y) <r} Show that B(x) is closed set…

A: Given: τ=B⊆X : ∀x0∈B, ∃rx0&gt;0, such that Brx0x0⊂B is a topology in X. Open ball Br(x)=y∈X : d(x,…

Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: an open ball in the set X, with its radius & satisfying -d(x, a), Vx € X at the open ball Be (x)…

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Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

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Q: If A and B are nonempty bounded subsets of R with A C B, prove that sup A < sup B.

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Q: 3. Consider the Cauchy-Schwarz inequality. That is, for u, v € R": Use some reasoning, or do some…

A: We have to prove ∑i=1nuivi≤∑i=1nui2∑j=1nvi2 Since we know the Holder's inequality…

Q: The set S=xER:x²-4<0} with the usual metric is . O A. Compact. B. connected. C. Not connected D.…

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Q: has

A: We have to choose the correct option from the given options according to the given information

Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

Q: Which of the following sets are convex and which are not? 2 a. {(x₁, x₂) x² + x²₂² ≥ 3}. b. {(x₁,…

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Q: (ii) Find a connected subset of R² which is not convex.

A: Note: Hi! Thank you for your question. As per the guideline we can solve one question at a time in…

Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: c) given an arbitrary convex function wcx) :R"→tR, prove fhat is a convex set.

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Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Show that the feasible set X = {r: g1(x) < 0,..., gr(x) <0} is a convex set, if all the functions 9;…

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Q: Show that both ℝ and ℝ2 may be covered by countably many open balls

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Q: 7. Let M = Q with the Euclidean metric. Determine whether the set C = {x € M: V2 < < v3} is open,…

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Q: 3. Prove if x is an isolated point of a set S, then x E bd S.

A: Since you have posted a multiple question according to guildlines I will solve first question(Q3)…

Q: Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Prove that any ball (open or closed or half open) in R³ is path connected.

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Q: Let E1, E2 be two compact subsets of RP. Use the Heine-Borel Theorem to prove that EU É, is also a…

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