MT 4800 – FUNCTIONAL ANALYSIS

Date & Time : 20-04-2006/FORENOON Dept. No. Max. : 100 Marks

ANSWER ALL QUESTIONS

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ii) If f Î X^*, prove that the null space Z(f) has deficiency 0 or 1 in X. Conversely, show that if Z is a subspace of X of deficiency 0 or 1, then there is an f Î X^* such that Z=Z(f).
b) iii) Show that every element of X/Y contains exactly one element of Z, where Y and Z are complementary subspaces of X.
iv) Let X and Y be normed linear spaces and let B(X, Y) denote the set of all bounded linear transformations from X into Y. Then prove that B(X,Y) is a normed linear space.

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v) Let X be a real vector space, p be a real valued function on X such that P(x+y) £ p(x) + p(y) and p(ax) = a p(x) ” x,y Î X and a ³ 0, and let Y be a subspace of X. If f is a linear functional on Y and f(x) £ p(x) ” x Î Y, prove that there is a linear functional F on X such that F(x)=f(x) ” x Î Y and F(x) £ p(x) ” x Î
a) i) If x ¹ 0 is an element of a real normed linear space X, then show that there exists an x Î x¢ such that x¢(x) = ||x|| and ||x¢|| = 1.

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ii) Let X and Y be Banach spaces and let T be a linear transformation of X into Y. Prove that if the graph of T is closed, then T is bounded. (8)

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b) iii) State and prove the uniform boundedness theorem.
iv) Give an example to show that uniform boundedness principle is not for every normed vector space. (10+7)

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v) Let X and Y be Banach spaces and if T is a continuous linear transformation of X onto Y, then prove that T is an open mapping. (17)
a) i) State and prove the Riesz – Representation Theorem.

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ii) If M and N are closed linear subspaces of a Hilbert space X and if P and Q are projections on M and N, then show that M ^N Û PQ = 0 Û QP=0 (8)
b) iii) If T is an operator on a Hilbert space X, then prove that T is normal iff its real and imaginary parts commute.
iv) Prove that how Hilbert spaces are isomorphic iff they have the same dimension. (7+10)

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v) If P is a projection on a closed linear space M of a Hilbert space X, prove that M is invariant under T Û TP =PTP
vi) If P₁, P₂, … P_n are projections on closed linear subspaces M₁, M₂, … M_n on X, then prove that P= P₁ + P₂ + …+P_n is a projection iff the P_i are pairwise orthogonal and in the case P is a projection on M=M₁+M₂+…+M_n. (5+12)
a) i) Prove that every element x in a Banech algebra A for which ||x–1|| < 1, is regular, and the inverse of such an element is given by .

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ii) Let A be a Banech algebra and x Î Then prove that the spectrum of x, s(x), is non-empty. (8)
b) iii) Let G be the set of regular elements in A and S be the set of singular elements in A. Prove that G is an open set and therefore S is a closed set.
iv) Show that the mapping x à x^–1 of G into G is continuous and is therefore a homeomorphism. (5+12)

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