1. Suppose that u, ve V and ||_ || = || v || = 1 and (u, v) = 1. Prove that u = v. Hint: Compute || u – v||2.
(4) w e suppose j is a measure, f E L1(μ),dom(f)-R, f 0 and EnaER: f(x) 2 n). a) Prove that limn-100 JE, fdy -0 (b) Prove that limn-0o n(En)-0 (4) w e suppose j is a measure, f E L1(μ),dom(f)-R, f 0 and EnaER: f(x) 2 n). a) Prove that limn-100 JE, fdy -0 (b) Prove that limn-0o n(En)-0
please help me,thanks! 3. Let Fo be a field with 9 elements. Consider the set S () e Fo] deg(f()) 18, f( f(1) (2)) (4) 0 and (a) Compute IS. (b) Prove that S is a vector space over F (c) Compute dimF, S Let V be a vector space over F. Prove that X C V is a subspace if and only if v, w E X implies av+wEX for every aEF 3. Let Fo be a field with...
1. let V be a vector space and T an operator on V (i.e., a linear map T: V--> V). Suppose that T^2 - 5T +6I = 0, where I is the identity operator and 0 stands for the zero operator ... Read Section 3.E and 3.F V) 1. Let V be a vector space and T an operator on V (i.e., a linear map T: V -» Suppose that T2 - 5T + 61 = 0, where I is...
(8) Suppose e, ,em) is an orthonormal list of vectors in V, and v E V. Prove that , e2 v, em)2 if and only if v Span(e.. .em) (8) Suppose e, ,em) is an orthonormal list of vectors in V, and v E V. Prove that , e2 v, em)2 if and only if v Span(e.. .em)
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo (12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
(3) 5. Suppose that f : D[0, 1] → D[0, 1] is holomorphic, prove that f'(2) < 1/(1 - 121) for all z e D[0,1].
(3) Suppose that f E C'((0, 1]). Given e > 0, prove that there exists a polynomial p such that lf-plloo -p'| <E (3) Suppose that f E C'((0, 1]). Given e > 0, prove that there exists a polynomial p such that lf-plloo -p'|
9.2.8. Suppose that E and V are subsets of Rwith E bounded, V open, and E CV. Prove that there is a C function f : E → R such that f(x) > 0 for x e E and f(x) = 0 for r&V.
(14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0 (14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0