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Algebra. Volume 1. Second Edition by P. M. Cohn

By P. M. Cohn

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Extra resources for Algebra. Volume 1. Second Edition

Example text

FF Ill< ___ ~_ _ =_rttllm _~ .. -JSIt .. _ _ __, .... ""_,,,,,,_•• __. ____ T ~ [X,yJ and X*, y* are the dual spaces to X, Y , . ~~· T, acting from Y* into (v) (R. A. 21) S~ LP(a,b) in N. DUNFORD, J. T. SCHWARTZ [lJ. •. ' ... ~"---_---'"'~ is precompact in and if and only if T* E K[y*,X*J. ) (N. DUNFORD, J. T. 7) The sequence {u} n c= ~ P < 00 u E LP(a,b) if and only for every measurable subset M ~ (a,b) . 2. ) v E W(a,b) , the mapping (8) uv 1 / p I un(t) dt is obviously an isometric isomorphism of LP(a,b;v) into LP(a,b) , and 1 p LP ' (a,b) into LP ' (a,b;v - ').

We will say that the imbedding and I is X is continuously (compactly) im­ X. 3) for I , Iu = u continuous (compact) or that the space 1 q 2 q x lu(x) eo. 13. Example. xC U --">. U n will denote the weak convergence of (see L. HoRMANDER [lJ, p. 182). The investigation of the validity of this u n to u (in x ). The symbol 72 73 _ ... 4) __ _. fF Ill< ___ ~_ _ =_rttllm _~ .. -JSIt .. _ _ __, .... ""_,,,,,,_•• __. ____ T ~ [X,yJ and X*, y* are the dual spaces to X, Y , . ~~· T, acting from Y* into (v) (R.

Then the operator - -'- 1 < Proof· The condition (i) A 00 L < 00 is equivalent to 1 L or i = H E [LP(a,b;v), Lq(a,b;w)] R . 33). x (ii) E; E (a,b) . 35) < ~ :i'; ~L (0 :i'; { f [ [f w( t ) a , l' , l' H E [L q (a,b;w -q ), LP (a,b;v -p )] R = ~~(E;;a,b,w,v,q,p) b [ JE; w(t) a a dt )1 I q [E;J ' r Iq' v 1- p (t) dt I v 1-p I (x) dx r r ) 1/p , (~') 1/r F (0 L , a which is the estimate (ii). -~-=~~---"~,-" "--------­ (iii) The proof is an analogue of that from part (ii). 7 imply the existence of numbers ~

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