By Louise Forsyth
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Extra resources for A Teacher's Guide to Imperialism: A History in Documents (Pages from History)
Y; y 0 ; Á; Á0 /dy 0 dÁ ¯ 0 dy d¯ Á: Proof. kCn/ /;j Cj˛jCjˇ j hÁi hyi for arbitrary j 2 N. lCl 0 /Cj˛jCjˇ jhÁim hyi ; where 2l > jmj C k and 2l 0 > j j C k. Moreover, we choose l; l 0 so large that 2l > m C n and 2l 0 > n C . 15). Rn Rn /. 9. 12. Rn /, where the integrals have to be understood as iterated integrals. 14. Let m; m0 2 R. Rn Rn Rn Rn / is the space of all smooth functions pW Rn Rn Rn S1;0 n R ! 1 C j 0 j/m j˛ 0 j 2 Rn for arbitrary ˛; ˇ; ˛ 0 ; ˇ 0 2 N0n. R2n R2n /. 15. Let m; m0 2 R and let p 2 S1;0 symbol.
10). x; / are estimated in the same way as before. 2 rigorous. Rn /. The proof is left to the reader as an exercise, cf. 58.
18) uniformly in x 2 Rn . 45. Rn I X/ ! Rn I X/ is a bounded linear mapping. Proof. 7. 46 (Inversion Formula). Rn I X/. x/ D F all x 2 Rn . Rn I X/ ! Rn I X/ is a linear isomorphism. 9 Final Remarks and Exercises Proof. 9 directly carries over to the vector-valued situation. 16). Plancherel’s theorem does not carry over to X-valued functions for a general Banach space X. 9) O /, respectively. x/, fO. / and g. Hilbert space. 47 (Plancherel’s Theorem). Let H be a complex Hilbert space. fO. /; g. Rn I H / !
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