By Hille E.

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**Extra resources for Note on the Behavior of Certain Power Series on the Circle of Convergence with Application to a Problem of Carleman**

**Example text**

22 E x a m p l e . An alternative (and in some ways more attractive) method of dealing with a differential equation is based on a direct attack on the equation itself and avoids the intermediate stage of constructing an integral 1. BANACH SPACES 16 equation. Spaces of differentiable functions may therefore be expected to play a role. Because of the limiting processes involved in defining derivatives, difficulties are encountered if R is an arbitrary set; this explains the restriction on R in the next definition.

21) is a Banach space. 23) are complete. 7) restricting also the derivatives. The following specimen result is enough for present purposes; its proof is easily deduced from the previous theorem. 10 Lemma. Let R be a n open subset of R", and consider functions f = ( f l , . . , f m ) mapping R into C". With the n o r m C1(R,Cm) i s a Banach space. 24 1. )] dx. lll. The C, spaces of which C1 is an example are a second class of Banach spaces important in applications. Because of technical difficulties connected with the inadequacy of the Riemann integral in this context, the discussion of these spaces is defcrred until the next chapter.

For K = {&} an orthonormal set in 3-1, the following hold: (i) Bessel's Inequality: for any f E 3-1, c I(f14n)I2 < Ilf 112. 2) (ii) (Best fit). 3) Proof. 2) follows on letting m m ---t 00. 4). 15 Lemma. Let K = (4,) be an orthonormal set, and suppose that (a,) is a sequence of scalars. Then the series Can& is convergent iff C la,12 is convergent. If the series is convergent, its sum, is independent of the order of its terms and Proof. 6) m=oo and it follows that if C Ia,I2 is convergent the sequence (C:=l is Cauchy, and since 3-1 is cornplctc, convergent.

### Note on the Behavior of Certain Power Series on the Circle of Convergence with Application to a Problem of Carleman by Hille E.

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