Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

Q: „Consider the space (R,T). where t={R,Ø}U{ACR:[0,1)CA} :Then (R,T) is not T, O T, but not T, O T2 O…

A:

Q: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

Q: a. Show that Ø, R E T. (by definition of Topology)

A:

Q: For the metric space (R, d) with d the usual Euclidean metric, which sets are in the topology T…

A: Wr Know that,"Let X = (-∞, ∞) and C consists of all intervals of the form (a,b).The arbitary union…

Q: Let d be metric on X, and de f ined di on X as di = max (d (r, ).0}, then dis semi metric but not…

A: Given d be an metric on X and d1 is defined on X by  d1=maxdx,y,0                     (i) To verify…

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

A:

Q: The Cartesian product X = X1 x X2 of two metric spaces (X1, d1) and (X2, d2) can be made into metric…

A: 4) To define a new metric on X which is different from the metrics mentioned in (1), (2), and (3).…

Q: Let X = [0, 1] and d be the usual metric, d(x, y) = |x- y| %3D O True O False

A: TRUE

Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded. %3D

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)…

A:

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.…

A:

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: Let X=R2 and let d be the Euclidean metric.  Define d3:  R2 x R2 to R by d3(x, y) = min{1, d(x, y)}…

A:

Q: Date. AC) Lat normed linear space X, Y= spomE and aex only if fla) = 0 wheneres f EX' omd f30…

A: To show the necessary part- To show that for each f∈X' whenever f vanishes on E, i.e.if f=0 then…

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D…

A:

Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

A:

Q: For every x, y E R, let p(x, y) = |x² – y²\. Is p a metric on R?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Let d: R x R R. defined as d (z, y) = |#1 - ml wherez= (21,2), y= (y.), then d is %3D Osemi metric…

A:

Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

A:

Q: \(d\) is a metric on \(X\). Show that \(\rho : X^2 \to [0,\infty)\) defined by \[ \rho(x,y) =…

A:

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: Let d: Rx R*→[0, ∞) such that d(XF)=√(x, y)² show that (R.d) metric space.

A: `

Q: Let d: Rx R R, defined as d(r, y) = 0, then d is %3D O not metric because d not sats fied d (x, y) =…

A: The correct option is the first option. i.e.  Not a metric because d does not satisfy d(x,y) = 0 iff…

Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

A:

Q: 2. Let (X, d) be a metric space and y E X. (i) Prove that the closed ball B[y,2] is a closed set in…

A:

Q: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

Q: å:RR→R be defined by dcx.y)=|xl+1yl, HxixR (R,d)is Than a metric space.

A: As per our guidelines we supposed to solve the very first question when the multiple questions are…

Q: Suppose d is Show that p: X?> [0,00) defined by a metric on X, p Cxy)= 3d (x,y) %3D 2+3d(x, u) is…

A:

Q: If X = {a, b} , how many metric can be de fined on X %3D infinite O none O only one O only finite

A: only one

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: Consider 2-dimensional hyperbolic space with metric ds = dr'+dy and y > 0. The length of a line…

A:

Q: If (X, d) is any metric space, show that another metric on X is defined by d(ry) d(r, y) 1+ d(x, y)…

A: The given problem is to show that X is bounded in new metric d(x,y)  Already given (X,d) is the…

Q: Consider a Z endowed with the metric if r = y d(r, y): 1979-" if r # y Where n is the largest…

A: According to the given information consider Z endowed with metric space:

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

A:

Q: Show that it A is subset of X any with the discre te metric then %3D

A:

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: Let d(x,y) = |1/x - 1/yl be a metric on RI(0}XRI(O}. Then the open ball of center 1/2 and radius 1/3…

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

Q: Show that, with z = Rw, for some bounded w, || z(jw)|| w llw(jw)| ||R||. = sup means that -

A:

Q: Verify that (S, d) is a metric space. 10.2 Prove that d(x, y) < d'(x, y)s/2 d(x, y) for all points'…

A: To prove dx,y≤d'x,y≤2dx,y for all points x,y∈ℝ×ℝ where d is the usual metric and d' is the…

Q: For any non empty set X, the discrete metro d:  X^2 to {0,1} is defined to by  d(x,y)={0, x=y and…

A:

Q: flz) = Im z is continuous %3D

A: The objective is to check f(z) =Im Z is continuous Or not.

Q: Show that a indeed perojective plane of gives 0rder BİBD (n+n+,n+),!

A: We need to prove that a projective plane of order n indeed gives a BIBD (n2+n+1, n+1, 1)

Q: Suppose that the tringle inequality is strict llel veetor s that means para V Qe R on non get 11 x+y…

A: Given: The triangle inequality is strict on non-parallel vectors, which means that, if x≠αy and y≠αx…

Q: Let (X,d)be full m.J,P, defined as D:Xx X R,plxy=min{1, d , is the full metric on X, show.…

A: Metric Space: Let X be a set. If a function d :X×X→ℝ satisfies the following properties (1) dx, y≥0…

Q: (X,d) be a metric space. Define f: X x X -> Real Numbers by f(x,y) = d(x,y) / 1+d(x,y) . show that f…

A:

Q: Let X = {2,3, 4,..} and de fined metric on X as d (r, y) = |- yl %3D %3D Find орen ball B (2,2) =…

A: Follow the definition: B(2,2) = { x in X : d(x,2) &lt; 2 }