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And ll xn - Yn ll < 4 - n for n = 1 , 2, . . In particular X and Y have isomorphic subspaces. 6. Prove the following. n )�= l consisting of polynomials. {b) There exists a sequence of polynomials which is a basis in all Lv [O, 1] , 1 � p < oo. 7. (a) Show that for I E C[O, 1] the Haar-Fourier expansion converges uniformly. {b) Construct a basis in C( �) . 8. Let (cpn(t))�= l be the Faber-Schauder system. Show that I = L:�= l an'Pn E LipQ [O, 1] , 0 < a < 1 if and only if l an l = O{n - Q ), so LipQ [O, 1] is isomorphic to £00 • 9.

Where Ll. is the Cantor set. X is Proof: Let B* be the closed unit ball of X* with w* -topology. A. 3) there exists a continuous map cp: Ll. � B*. ) by T: X ---+ T (x ) ( ) = t5 cp x ( t5 ) ( ) for t5 E Ll.. a It is easy to check that it is an isometric embedding. ) by C[O, 1] in Theorem 4. ) into C[O, 1] . ) we assign F E C[O, 1] such that F I Ll. = I and F is linear on each interval of [O , 1] \ Ll.. (xn)�=l x X, X of a Banach space is called 5. B. Isomorphisms, Bases, Projections §6. )� E: a unique sequence of scalars ( an = l such that x = = l anXn· Let us emphasize that the sign '=' means here that the series = l anXn converges to x in the norm of X.

And combination of 's has norm 1 . l ! (H00 , (Ld H1 ))­ topology. = Let T: H00 (D) ---+ A(D) be defined by Tl ( z) 1( � ) . (H00 , (Ll / H1 ) )-topology into the norm topology. = l an l (cpn)�=l (an)�= l (qn)�=l qn+ l 3qn · = + an qnt + �Pn)) (Riesz products) Let be a sequence of real numbers such that ::; 1 for n 1 , 2, . . L. Show also that fi,(qn) = ei'l'n an. Suppose that ( an ) �� oo is a sequence of numbers such that lim infN-+oo II L �N aneinOI I < Show for some p, 1 p < P that if 1 < p < then there exists a function I E Lp(Y) such that }(n) = an for n = 0, ±1, ±2, .

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A Concrete Approach to Abstract Algebra by W. W. Sawyer


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