let P and of [a, b] of be el de are [a, b] show bounded function two ary that [[F, Q] < U[F₂P] on [a,b], Let measurable partitions.

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: 3. Let X and Y be normed spaces and f:X→Y be a bijection linear function. prove or disprove f is…

A: It is true that if f is a bijection linear function from a normed space X to a normed space Y, then…

Q: Exercise 3 Determine all possible values of a € R such that the function f(x) = e* -ax³ be convex on…

A: Please see the below picture for detailed solution.

Q: Theorem 4. Let X be a set, and let F: X Y be a l-1 and onto function. (a) F-lo F = Idx (b) FoF- =…

A:

Q: Let f : A → B be a function. Show that the set P = {f ^b | b ∈ B} is a partition of A.

A:

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

A:

Q: In Exercises 1-7, (X,M) is a measurable space. 1. Let f : X R and Y = f-1(R). Then f is measurable…

A: Given: Let's consider X,M is a measurable space.  Here f:X→ℝ¯ and Y=f-1ℝ

Q: (C0]) }

A: The statement is false. Consider f:[0,1]→ℝ defined as f(x)=tan-1πx2   x∈[0,1)0              x=1

Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

A: Let f:X,Tx→Y,Ty be a function and a is open in Tx. To prove that the function is continues at a.…

Q: 6. If f is bounded above function on [a, b] and let P be a partition of [a, b), then A. L(f, P)…

A: Given :

Q: 4. Suppose X and Y are normed spaces, dim Y <0, f:X→Y is linear and f(x) = Y. If ker(f) is closed.…

A: To prove that ƒ is continuous, we need to show that for any given ε &gt; 0, there exists a δ &gt; 0…

Q: Prove the Cauchy-Schwarz Inequality, ∣u ∙ v∣ ≤ ∣ u∣∣  ∣ v ∣∣.

A:

Q: 13. Let X be a set, let (Y, Ty) be a topological space, and let f: X→Y be a function between sets.…

A:

Q: Prove that the function g: R &gt; R defined by f(x) = 1+2^x is one-to-one but not onto.

A: Consider the function f:R-&gt;R defined by:

Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: Prove that if R: R -> R is continuous, theis b2n the image of a bounded set over f is bounded

A: As per the question we have to prove that if any function f : ℝ → ℝ is continuous, then the image of…

Q: الموضوع show that if foret A frix by and ; YーZ is a continuous map then gFo relA gf,:X b Z

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Q: Q5. (a) Find the value of A, for which the following differential equation will be an exact DE: (xy'…

A: In the given question we have to find the value of the for which the differential equation   is…

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 U and y be bounded sets. if u er i tre M+ (V) ≤M₂ (V) and M^ (U) SM³ (V) 1

A: Given :  U and V are bounded sets with  U⊆ V. To show :  m*(U) ≤ m*V  and  m*U ≤ m*V,  where m*U and…

Q: Suppose f ond 9 are continuous functions on [0. b] ond thand gee)>0 for all x E [0,b]. Prove that…

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Q: (3) For any subsets A, B C R" we denote by AAB = (A\ B) U (B \ A). Let f, f1, f2,·. E L' (R") such…

A: Consider the given functions f1, f2,...∈L-1ℝn such that: limk→∞∫ℝnfk-f=0 and A1, A2,...⊂ℝn  be…

Q: is an Let V(F) be an inner product space. If 0 + o e V then prove that orthonormal set.

A:

Q: Prove that if f is a constant function on [a, b] then f = S(f,P) for any tagged partition P of [a,…

A: This is a question of R integration.

Q: Please help me and show your work on paper Thank you

A: The first part of the question is asking us to prove that the sum of the kth powers of the first n…

Q: 7. If in an inner product space, (x, u) =(x, v) for all x, show that u = v.

A:

Q: Let f(x, y, z) = x³ – yz?. Find af af of and ax' dy az

A: Given : fx,y,z = x3-yz2

Q: Construct a one-to-on continuous mapping of the set {(#1, 12, - ..., In); 0 < T; < 1. j= 1, 2,…

A: Given that : x1,x2,x3,.........,xn, 0&lt;xj&lt;1, j=1,2,3,.....,n So we have to take ,…

Q: on 1: a) Let O be the set of odd numbers and O' = {1,5, 9, 13, 17, ...} be its subset. Define the…

A: Since the part b is independent of the first part so I am answering the first question only for the…

Q: (b) Show that xE R, x2 - 4x + 5 > 0.

A: As this is a multiple question type, only first question is to be solved, according to the Bartleby…

Q: Let x ∈ R ^n and r > 0. Is the set B(x, r) = {y ∈ R^ n : ||y − x|| ≤ r} a closed set in R ^n? Y/N

A:

Q: Let f: A→ B, where A= {1,-1}, B= {1,2,3}, and f(x)= 3×x²+2, find Range إجابتك * Codomain إجابك * Is…

A:

Q: 3. Describe the domain and range (i.e., comment on whether the domain and range are…

A: Consider the function, fx,y=lnx2+y2 To determine the domain and range of fx, y. The function is not…

Q: 4. Find x and y such that =x'y subject to *+*= 36, x20, y20 attains a maximum value. A) (24,12) B)…

A: Given: x+y=36, x≥0, y≥0   ……1   To find: x and y when z=x2y is maximum.

Q: Prove: f € R(a) if and only if for each ɛ > 0, there exists a partition of [a, b ] , say go, such…

A: f∈Rα if and only if for each ε&gt;0,there exists a partition of a, b,say P, such that UP, f, α-LP,…

Q: 5. Assume that f:Z+→Z*. Then the type of the function f(x) = LxJ +2 is considered a) Onto function…

A: