Let f be a bounded function on [a, b]. If P and Q are partitions of [a, b], prove that L( f , P) ≤ U( f , Q).

Q: Show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2,…

A: We have to show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v…

Q: V [2] Let u* be an outer measure on a set X such that B is a subset of X, then u*(AU B) = u*(B).…

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Q: Let A, B C R be bounded below and let A+B = {a+b|a EA, b E B}. Is it true that inf(A + B) = inf(A) +…

A: Let  be bounded below and let .To Find:Is it true that .

Q: Show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3).…

A: An inner product on real spaces V is a function that associates a number u, v with each pairs of…

Q: Show f e Lla, b] if and only if |f| E L[a,b].

A: The infinity norm of a function f on a,b is defined as: f∞=infk:fx≤k, almost every where on a,b.…

Q: Let f: [a, b] → R be a bounded function, and let P be a partition of [a, b]. Let P be a refinement…

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Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: What does it mean to say f : (a,b) →R is a bounded function? If a < b in R and f : (a,b) → R is…

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Q: 2. Let X1 and X2 be sets. Define functions T1 : X1 × X2 → X1, T2 : X1 × X2 → X2 by T| (X1, X2) = x1,…

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Q: Let S be the set S = {a, b, c, d} and consider the function f : P(S) → Z given by f(X) = |X| Show…

A: Let S be the set S = {a, b, c, d} and consider the function f : PS → ℤ given by f(X) = |X| Where X…

Q: A = {2, 3, 4, 5} and B et f : A → B be a func ow. Which of the following

A: A function f: A → B is said to be an ONTO function if, The range that exists for f is the set B…

Q: Let A = (1, 2, 3, 4, 5) and B = {a, b, c) and f = {(1, a), (2, a), (3, b), (4, b). (5, b)}. Choose…

A: Given. (a) f is a onto function because image of 1 and 2 are same also image of 3, 4 , 5 is same.…

Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

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Q: Let X = {o, p, s, w}, Y = {a,b, e, g, k}, and Z = {1, 2, 3, 5, 7, 9}. Let ƒ : X → Y where f = {(o,…

A: X=o,p,s,wY=a,b,e,g,kZ=1,2,3,5,7,9f:X→Yf=o,a,p,g,s,e,w,bg:Y→Zg=g,3,e,2,b,9,k,7,a,1

Q: Let f,g : R → R be bounded and uniformly continuous. Prove that fg is also bounded and iniformly…

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Q: f (UMEGM) = UME¢ f (M)where C is a nonempty collection of subsets of C.

A: Set theory

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

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Q: Prove that if f : R → R is continuous, then the image of a bounded set over f is bounded.

A: Given f:R-&gt;R is continuous function  Let A is bounded subset of R  So the limit point set A' is…

Q: Let g : X c R"→ R be a function defined on the finite set X. Does g achieve a maximum and/or a…

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Q: Ja (16) Let f e Lla, b]. Show that if g is a bounded measurable function then fa E Cla bl

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Q: Let f: X→Y and g: Y→Z be functions. Prove that if f is an onto function and g∘f is a one-to-one…

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Q: Prove that the following relation from Z= {c, o, u, n, t, r, y} to X= {h, a, z, k}is a onto but not…

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Q: Let g: [a, b] → R be a bounded function, and suppose that there exists a partition P so that L(g, P)…

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Q: Let P denote the set of functions f in C([0,1]) with f(x) > 0 for all x. Then P is a closed subset…

A: Given that, P denotes the set of functions f in C0,1

Q: Let A = {a, b, c, d, e}, B = {1, 2, 3, 4, 5} and let us consider the function f : A → B, f(a) = 3,…

A: Given that:  f : A → B, f(a) = 3, f(b) = 4, f(c) = 2, f(d) = 5, f(e) = 1.   Since, the images (i.e.…

Q: g) for any bounded

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Q: Let f : Rn → R be convex with dom(f) = Ω ⊂ Rn an open set. Show that f is a continuous function on Ω

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Q: Let a, b, c, d ∈F. Prove that the function δ: M2×2(F) →F defined by δ(A) =…

A: Let a,b,c,d belongs to F.Then it is required to prove that the function is linear function.

Q: ) Let f be a continuous function that maps a finite closed interval [a, b] into itself. Prove that f…

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Q: Let f be the subset of Z × Z defined by  f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z?…

A: Let f be the subset of Z × Z defined by  f = {(ab, a + b) : a, b ∈ Z}. We observe that  That is the…

Q: (iii) (f/g)(x) = g(x) if g(x) #0 for all z € M. Prove that cf. fg and (f/g) are all continuous. 8.…

A: To Prove: Restriction of a continuous function is also continuous on Metric space.

Q: 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: Given that F is a bounded function on the interval a,b and P and Q are the two measurable partitions…

Q: If f is a bijection of A onto B, show that f-1 is a bijection of B onto A.

A: Since  is a bijection from  onto , the inverse  is a bijection from .

Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

Q: Suppose that f:[a,b] → R is a bounded function for which there sa partition P of [a, b] with L(f,P)…

A: Given the statement Suppose that f: [a, b] → R is a bounded function for which there is a partition…

Q: Let A be a nonempty set and let f: A -> A be a function. Prove that if fof = iA, then f is…

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Q: Let f be a real-valued function which is bounded on (a, 6), P be a partition of (a, b] and sup{f(r)…

A: