a. In (R, d) where d(x, y) = |x – y| (Euclidean Metric space) show that: Q (The set of all rational numbers) is dense (Q = R).

Q: What is the global minimum of the function, x² 5 f(x) 3 + 2 on the interval [-2, 2] ? a) 2 b) 23 3…

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Q: Suppose v1, V2,..., Vm E V. Prove that {v1, v2, ..., Um} = (span(v1, v2, ..., Um))-.

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Q: lim (Vx + 45 V:

A: Given,          limx→∞x+45-x

Q: 18. Determine whether each of these functions is a bijec- tion from R to R. a) f(x) = -3x + 4 b)…

A: As per guidelines, we will solve first three subparts of the question. a) Given function is fx=-3x+4…

Q: let M be the lebseque measure on B IR) Calcutate (1) Mes), where S = Where S= { Yn t h+3 i) M( n…

A: We know that measure of countable set is '0'

Q: Show that every countable subset of R" has Hausdorff dimension 0.

A: claim- to show that every countable subset of ℝn has Hausdorff dimension 0. it is known that…

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Q: Let f(x, y) = (2x - y,x-2y); (x, y) E R x R, (R is set of real numbers.) %3D a) Show that f is one…

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Q: positive real numbers such tha ith proof the minimum value of

A: Given, x,y,z be a positive real number such that x4+y4+z4=1.The minimum value of…

Q: Submit through Canvas as indicated. T u = (2,-5) and v= −3i+7j

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Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

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Q: Let X={a, b, c, d, e} and let T={X, non zero, {a}, {a, d}, {a, e}, then (X, T) is neither Hausdorff…

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Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

A: Given that X be an inner product space and x,y∈X Let, x⊥y. The objective is to prove that…

Q: State true or false with brief justification: L1[a,b] is a reflexive space

A: We use the Theorem: Let X be a separable normed space. If the dual space X* of X is non-separable,…

Q: (R,t) usuall top space and A=(0,1),B = (a,b); a<b. prove A=~B ??

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Q: Let V = R³ and W = {(a, b, c) E V | a² + b² = c?}. Is W a sub- space of V? If so, what is its…

A: As per the question we are given the three dimensional vector space V = ℝ3 and a subset W is defined…

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: 4={(-21.r):reR} II) B=(2r.-31): te R} III) C={(2r+3,-41) re R} I)

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Q: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

A: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

Q: Let V is an inner product space. Prove, Va, y EV ||x+y||² + ||x − y||² = 2||x||² + 2||y||* . How…

A: To prove a result of the inner product space.

Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: If P and Q are represented by the numbers z1 and z2 such that H- then the circumcentre of AOPQ,…

A: Given P an Q are represented by the numbers z1 and z2 such that |1z1+1z2|=|1z1-1z2| We need to find…

Q: Prove or disprove the following statement: A) Every T'3-space is T4-space?( explain your answer)

A: The statement "Every T3-space is a T4-space" is true. To understand why, let's first define the…

Q: Find the set the units of the integral domain Z [√5] = {a+b√-5: a, b € Z}.

A: Find the set of the units of the integral domain .

Q: ": Prove that ( R", ||||) is complete metric space.

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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: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

Q: Let v= C[-1,1] What is the norm of the functionf(x) = - x in this space? O A) 2V5 O B) 2V10/5 OC)…

A: An inner product in the vector space of continuous functions in [-1, 1], denoted asV=C-1, 1, is…

Q: Show that 7(u, v)= (²-²,2ue) maps the triangle-((u, v):05≤SS 3) to the domain D bounded by z = 0, y…

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Q: iü) #lt)= - N2z+S b) Compute Re fli) when: $(2)= √2-₁² and f(0)= -i Which of the following functions…

A: Analytic function : A function f is analytic at a point complex point z=a iff the f(a) is a finite…