The space l3 is a Hilbert space
False. The space \(\ell^3\) is not a Hilbert space. In functional _______ __________ _______ _____ __________ _______ ____ ________ _________ _____.
________ _____ _____ ____ ____ _________ __________ ___ _____ ___.
________ ________ _________ _______ _____ ___ _______ __________ _____ _________ ___ ____.
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Let X be a vector space. Let be two norms on .X When are these norms said to be equivalent? Justify your answer.
Let . X = For x=(x1,x2,x3)
Let
Show that and
are equivalent.
Let } u{ n be the sequence in l2 with 1 in the th n place and zeroes else where prove that the set } u{ n is an orthonormal basis for l 2
Check whether the identity map on an infinite dimensional space is compact.
Characterise all bounded linear functionals on a Hilbert space.
Let T be a linear map defined on X by
Show that T is a bounded linear map such that T = .1
Let ]1,0[C X = with Sup norm defined by f = Sup{| )x(f |}. ]1
Let ]1,0[ X = C′ and ] Y = 11,0[C and let T : X → Y be the linear operator from X to Y given by ,f )f(T = ′ the derivative of f on ].1,0[ Show that T is not continuous.