• Exercise 3 Prove that if S and T are both bounded linear operators from the normed space X to the normed space Y, then for all a E C and BE C the operator H defined by ÎĤ = aŜ + BÎ a) is a linear operator. b) is a bounded operator.
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Q: he set V = A {(x, y) ∈ R² | x> 0 and y> 0 |} is a vector space
A: False
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Q: 3c
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A: Please check step 2 and 3 for solution.! We will verify axioms of vector space.
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- No hand written solution and no imageLet A be a bounded linear operator on a Hilbert space H and ||A(x)||=||x|| ¥xe H. Show that A is unitary.7. Let V be the vector space of polynomials in two variables x and y of degree at most two: V = {ax² + bxy + cy² + dx + ey + ƒ | a, b, c, d, e, ƒ € R}. Let T be the linear operator on V defined by T(g(x, y)) = Ə Ix 9 (x, y) + Find the Jordan canonical form of T. Ə dy I (x, y). ду
- Let N and N' be normed linear spaces with the some scalars and let T:N-N' be a linear T Transformation then prove that T is bounded if and T only if T is continuous.(b) Let X bę a normed linear space. Let Вх, г) %3D (у є Х: || x-у ||Plz answer correctly asapQ₂ Prove that such as ( x ₁ y ) + ( x ²₁ y ²) = (x + x ² + 1₂ y + y ² + ₁₂ and K(x, y) = (kxky) is not a vector space V = set of all pairs of real numbers. (x, y)Prove disprove that (R,T co-finite) is Te-space or1)LetT(x, y, z) := (2x + y, x + y + z, y – 32) de R³ in R3be a linear operator. Considering the usual internal product in R3: a)Show that T is a self-adjoint operator but is not orthogonal. b}lf V = (2, –1,5)andw = (3,0, 1), · Make sure that (Tv, w) = (v,T'w). c) Display a base of eigenvectors and eigenvalues ofT and verify that it is an orthogonal base. From this base, find an orthonormal base.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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