12. Suppose that a, b E R and a < b. Show that the vector space C[a, b] of all continuous complex valued functions defined on [a, b], with supremum norm is a Banach space. ||f||= sup f(t)\, te [a,b]

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Q: Prove that the set S = {(x1, x2) | X1, X2 € R, x1 ≥ 0, x2 ≥ 0} is not a real vector space.

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Q: 3. Prove that if r, y E RP, then ||r + yl|2 = ||x||2 + |ly||² holds if and only if r y = 0. 4. Prove…

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Q: 1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 +…

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Q: 9. Which of the following subsets of the space of real valued functions on R are vector spaces?…

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Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V x V → V…

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Q: Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤…

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Q: Let V be a vector space and suppose there exist maps S, T = L(V, V) such that range(S) ℃ null(T).…

A: Let V be a vector space and suppose there exist maps S,T:V→V such that .We need to prove that the…

Q: 4. Show that the norm on l0 given by I| (11, .., In...) |lo= sup{|r;| : j= 1,2, ...} can not be…

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Q: ) For the vector space C[0,1], define the inner product (f,g) = , f(x)g(x)dx. Let f = x + 1 and g =…

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Q: 2.53 Let B[a, b] denote the set of all bounded functions on [a, b). Is Bla, b] a vector space? Prove…

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Q: The set Fun(R) of all eal-valued functions defined on R. together with the standard function…

A: Definition of Real-Valued Function:                       A real-valued function is a function whose…