Let ⁄2 be a bounded interval. For any g € L²(2), define the operator T, by T,(f) = f Then T, is a linear functional on L²(N). 22

Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

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Q: A = {1,2,3}, B = {4,5,6}, C = {7,8,9,10} define the functions f :A→C and g:B→C by f(1) = 7, f(2) =…

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Q: Suppose that g: [a, b] R is a continuous function. Show that there exists a point c € (a, b) so that…

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Q: 3. Let L be a linear operator on R and let L(1) = a . Show that L(x) = ax for all x E R .

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Q: Every bounded continuous function attains its maximum and minimum values.

A: Here’s a sketch of one possible argument. Let u=supx∈[a,b]f(x). (Note that I allow the possibility…

Q: Exercise 1. Let f(x) be the function 2х — 1 if x 2. - (a) Find the value of c that makes f(x)…

A: Given f(x)=2x-1if x<-1cxif -1≤x≤2x-2if x>2

Q: Exercise 1. Let f : A → B be a function with X a subset of A. Prove that if f(X) = 0, then X = 0.

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Q: 5. Let f: Z → Z be defined by f(n) = n² + 1. (a). Is f a one-to-one function? Explain. (b). Is f an…

A: Given- The function is f(n)=n2+1 and f:ℤ→ℤ. Definition- Let a function from X to Y, One-to-one: If…

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: 4. Define the operators T and U on the space P(R) of all polynomials in variable x as follows:…

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Q: Theorem 18.1 states that "Let f be a continuous real-valued function on a closed interval [a, b].…

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Q: q17. Let f, g, h be continuous functions on an interval [a, b]. If f is orthogonal to g and to h,…

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Q: (a) Prove that if m ₹ N, then the function f(x) = xm is continuous on R.

A: We have to prove that, the function f(x) = xm,   m∈ℕ  is continuous on ℝ.

Q: Show that the function f: N² → Z given by f(m, n) = m²-n² is neither injective: suriective. nor

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