non-empty set and d:X x X R de fined by d(x,y)=31 x- 41_ 2+|xl+lyl or disprove that d is a metric on X Prove

Q: Let (X,d) be a complete metric space with X not ∅. Suppose the function f : X → X has the property…

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Q: Let X be an infinite set and let d be the discrete metric on X: {: 0if p=q 1 if p + q. d(p, q) = (a)…

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Q: G Let d. R&Rしえ be dedned by des,)ー e Than (lR, d) is a metvic space. O Let diRxR-R be dehned…

A: Given

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

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Q: Let (R. d) be discrete metric space then R is not complete O True O Faise

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Q: O:- (Distance between sets) The distance DUA, B) between two nonempty Subsets A and B of metric.…

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Q: f(x,y) = (In(y + x²)) x² + y² D = {(x,y) ERIy2x²-1 and y ≤4-e-*) (b) Observe the set Is D…

A: (b) The given set is D=x,y∈ℝ : y≥x2-1 and y≤4-e-x To find: The given set is bounded or closed or…

Q: Q2:(a) In Euclidean metric space (R, 1. 1),if A = {y € R: y = 2 cos (2x); x € R} then fined the…

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Q: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

A: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

Q: Let X be a metric space with metric d. Let r > 0 and define the open ball center at IE X with radius…

A: here we only need to show all the condition of topology.

Q: Let X = Rand def ine |-|| on X as ||r||= V then X is not normed linear space O True False

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Q: 5. Let d be a metric on X. Determine all constants k such that (i) kd, (ii) d+ k is a metric on X.

A: Given that d be a metric on X. Then d:X×X→[0,∞) satisfies the following conditions for x,y,z∈X, (a)…

Q: Define p: R^2 x R^2 to [0, infinity) by the rule p(x1,y1),(x2,y2)=squareroot((x1-x2)^2+(y1-y2)^2)…

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Q: Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i.…

A: Given metric is          d((x1, x2, x3), (y1, y2, y3)) = |x1 - y1| + |x2 - y2| + |x3 - y3|.   For…

Q: Let (X, T) be a topological space and A,B CX- ConstYuct the general Yelation between Fr (IntlA) and…

A: For any subset B of a topological space X , Frontier or boundary of B is defined as  Fr(B) = CL(B) -…

Q: 3. Let Ac R' where A = {x: x 1}. Describe the quotient space topology on R'/A obtained by…

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Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: A normed space X is said to be strictly convex if for x # y in X with |||| = 1 = ||yl|, we have…

A: It is given that X is normed space and for x≠y in X with x=y= 1, then  x+y<2. We have to explain…

Q: Let X be a metric space with metric d. Let r > 0 and define the open ball center at IEX with radius…

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Q: Let p1 and p2 be points in a metric space X. Prove that the set p1 ⋃ p2 is compact in X. Now, let…

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Q: 3. Let d(x, y) := |2 - 2| for x, y E R. . is a metric on R. Show that (R, d) is not a complete…

A: Cauchy sequence: A sequence x1, x2, x3, ... in a metric space (X, d) is called Cauchy if for every…

Q: Set (Q, M, µ) a space of o-finite measure. we define the aplication 4: L° (Q) → L'(Q)* given by =…

A: Let g,h∈L∞(Ω and α∈F Claim:- ϕ(αg+h)=αφ(g)+φ(h)

Q: A metric space (X,d) such that the function d1(p, q) = (d(p, q)) i not a metric in X

A: Since (X, d) is a metric space. So, d is a function defined as d : X×X→R+∪0 and satisfy the…

Q: Let X = Rand def ine ||.|| on X as ||x|| = 1 %3D then X is not normed linear space True O False O

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Q: (b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact that ||ax + by |…

A: Given that, Euclidean metric d on Rn is defined as d(x,y) = || x - y || for x, y in Rn Part(a): Let…

Q: et X be a metric space with metric d. Let r > 0 and define the open ball center at EX with radius r…

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Q: 1. Let, y eR. Determine if the following are metrics on R or not (a) d, (x, y) = /1x - y| (b)…

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

Q: A. Let H be the set of all points (x, y) in R2 such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

Q: (йх'ў):(wx2)

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Q: Let X be the subset (R x {0}) U (R x {0}) CR2 Show that the quotient space X/~ is locally Eue line…

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Q: Q3: state whether the following statement are True or False : 1. The set A = {x E R:x2-5x +6 <0} has…

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Q: 4. Let X and Y be metric spaces and f : X → Y be continuous. Let SC X. Show: (b) If S is compact,…

A: First, we prove that fS¯⊆f(S)¯. Given that f is continuous and S¯ is compact, we can say f-1Y-f(S)¯…

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1, 1] with the usual…

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Q: roblem 3 Let (X, M,µ) be a space with measure. Let and A,B E M µ (B)=0. Prove that

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Q: 3. i) Let (X, d) be a metric space. If æn → x in (X, d) and y e X, prove that lim d(xn, y) = d (x,…

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Q: Q If X R, define the function ||- ||: R →R*u {o} by ||-|| = |x| + 1. Is (R, ||-||) normed space.…

A: Solution: We will use the following definition: The function .:ℝ→ℝ is a norm if it satisfies the…

Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+ z) = d(x,y), x,y € N…

A: Answer

Q: Show that the following map on R", d2(x,y) = E-1 |Xi-yil ;), x,y ER". =1\2i 1+|xi-Yil Is a metric…

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Q: If X =R and d(x,y)= 1, then X not metric semi metric metric space

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Q: 6. Let A and B be two connected subsets of X and An B + 4. Show that AUB is connected. In…

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Q: Let (X, d) be a metric space. Define dˆ : X × X → R, by: ˆ
d(x, y) = min{1, d(x, y)}. (a) Prove that…

A: To verify that d^ satisfies the axioms of a distance function

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1,1] with the usual…

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Q: This is a real analysis question. Let (X,d) be a complete metric space with X not ∅. Suppose the…

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Q: (c) Given the set T := {(x, y) = R²: |xy| ≤ 1}. Is T a compact set? Show your working. If you say it…

A: Given c) T=x,y∈R2:xy≤1d) Given a metric space X,ρ and two compact subsets S,T∈X

Q: EC. Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric). i. Explain…

A: We will use facts from general topology to solve this question.   Note that we know that [a,b] and…

Q: (Product of metric spaces) spaces (X1, d1) and (X2, d2) can be made into a metric space (X, d) in…

A: Given that

Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y + z) = d(x, y), x,y E N…

A: Given: N be a linear space and dbe a metric on N such that d(x+z, y+z)=d(x, y),  x, y ∈ N and  d(ax,…

Q: Exercise 1.33. Let (X,d) be a metric spáce and d(x,y) 1+d(x,y)' d* (x,y) = Prove that the family of…

A: This is problem from my old works, where I used e instead of d*.

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.