By Lucia M., Magtone R., Zhou H.-S.

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**Example text**

That f(a) does not depend on the choice of the punched disc envelope E is proved in the next theorem, which is the main result of this section. 33 § 7. A Functional Calculus for a Single Banach Algebra Element Theorem 4. (i) Given f e H (D), f(a) is independent of the choice of the punched disc envelope E for (Sp(a),D). (ii) The mapping f ->f(a) is a homomorphism of H(D) into A that extends the natural homomorphism f -> f(a) of R(D) into A. (iii) Given a compact neighbourhood K of Sp(a) contained in D, the mapping f -> f(a) is continuous with respect to uniform convergence on K.

Then Sp(sinh)cU(O, 1), and arc sin(sinh) = h . Proof. We have Sp(sinh) = {sinA: AESp(h)}, cJ-1, 1[cU(0, 1). Therefore and so Sp(sinh) rtJ arcsin(sinh)= L: IX. (sin h)· . n=1 Let D be an open neighbourhood of Sp(h) such that sinzE U(O, 1)(ZED). Then arc sin(sinz) is a holomorphic function of z on D, and, since arc sin (sin t) = t (tE [ -nI2, nI2J), we have arcsin(sinz)=z (zED). 57 § 11. Approximate Identities D= Let s,,(z)= 1 IXk(sin4. Then s,,(z)--+z uniformly on a neighbourhood of Sp(h). e. h=arcsin(sinh).

Since the interior of a finite intersection of sets is the intersection of their interiors, we have n V(C,p)· m KcintE=U((o,po)n j= 1 Proposition 3. Let K, D be subsets of