Let f,g ∈ L∞[a, b] and c ∈ R. a) Show that cf ∈ L∞[a, b]. b) Show that f + g ∈ L∞[a, b].

Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: Let f : R R with f(x) = -2x +5 Prove that f is onto.

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Q: Determine if f : Z × Z → Z is a surjective function if: а) f(m, п) — 2m — п b) f(m, п) — т? — п? -…

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Q: Suppose f : N → N has the rule f(n) = 4n + 1. i) Determine whether f is 1-1. Justify your answer…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: ス) For each Funci injective, surjectie, both, ar neither. Give short justi ficatiums for you la) A=…

A: 1) f(x) = x2 for injective take two elements x1 and x2   from set A  f(x1) =f(x2)⇒x21=x22⇒x1=x2  and…

Q: 4. Show whether or not f is bijective. Letf : R → R_be defined as f(x) = I피+1 %3D

A: A function f is said to bijective if it is one-one and onto

Q: Show that in R with the usual topology, the only open sets connected by trajectories are the open…

A: Given that R with the usual topology  A collection of of subsets of R which can be expressed as a…

Q: Let T: R3 -> R2 be defined by T(a, b, c) = (a + b, 2a —- c). DescribeT-1(1, 11).

A: T: R3 -> R2 Hence, T-1 : R2 ----> R3 So, necessarily we have to find a, b & c such that…

Q: Let f: R2 R2 be defined f(x, y) = (3x + y,x+y). (a) Prove that f is a bijection. (b) Keep f the same…

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Q: Forz € C, define the linear operator T₂ : C → C by T₂(w) = W. Characterize those z for which Tz,…

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Q: Let fi(x) E Fun (R,R) be the constant function I, wheret ficx)=1 for all xG RRe Prove that S { sec?…

A: Given: fx∈funℝ, ℝ is a constant function. such that: fx=1……1 S=sec2x, tan2x, fx

Q: Sketch the domain D in the (x, y)-plane satisfying ≤ y ≤ 2e- and e ≤ y ≤ 3e. In your sketch, label D…

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Q: Consider the relation R = {(x,x): x € Z} on Z. Is R reflexive? Symmetric? Transi- tive? If a…

A: We have to check given relation R for reflexive,symmetric and transitive. Concept required: R:A→B is…

Q: Let f = u + iv be analytic in the connected open set and assume u and v satisfy u? + 3v2 = 4. Show f…

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Q: Exercise 3. Prove that a relation R on A is antisymmetric if and only if RNR-' C {(a, a) | a E A

A: We know that , A relation R on set A is antisymmetric  if either x , y∉ R  or (y , x) ∉ R whenever…

Q: nach limits to show that the space ℓ∞ is not reflexive.

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Q: Let f be a function that takes elements of R?rR? into R' as follows: f(u, v) = |u1v1|+ |u2v2|, where…

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Q: Gragh Hhe funchans fQ)=} and of)= 3X-1 2 X-L a) White a/2) as a transfarmation of flc) : g= f(xta) +…

A: Given: fx=2x and gx=3x-1x-1 To Find: i. Write gx as a transformation of fx:gx=fx+a+b.ii. Describe…