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By Lynn H. Loomis

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ThE follow1ng are examples of important types of Banach algebras with which we shall be less concerned. 4. The complex-valued functions continuous on the closed unit circle II z \1 ;;;! 1 and analytic on its interior II z II < 1, under the uniform norm. 5. ,. + § 19. FUNCTION ALGEBRAS The imbedding of a Banach space X in its second conjugate space X**, that is, the representation of X as a space of linear functionals over the Banach space X*, is an important device in the general theory of Banach spaces.

INTEGRATION 43 If the domain ofF is restricted to the subspace of LP consisting of the functions which vanish off a given summable set Sx, then the norm ofF is decreased if anything. The preceding theorem furnishes a function/1 defined on s1. If s2 and/2 are a second such pair, then the uniqueness part of lSC shows that / 1 and /2 are equivalent on S 1 n S2. Suppose now that p > 1. Let b = lub Ill/ II q: all such/} and letS.. be a nested increasing sequence such that II f,. llq j b. , with II /o llq = b.

For Ax= 0 if and only if (Ax,y) = 0 for ally E: H, and since (Ax,y) = (x, A*y), this is clearly equivalent to x being orthogonal to the range of A*. liB. Theorem. The involution operation T ---+ T* has the following properties: 1) T** = T 2) (S + T)* = S* + T* 3) (~T)* = ~T* 4) (ST) * = T*S* 5) II T*TII =II Tll 2 6) (I+ T*T)- 1 E: ffi(H), where I is the identity transformation. Proof. Properties 1) to 4) are more or less obvious from (1). For instance 4) follows from (x, (ST)*y) = (STx,y) = (Tx, S*y) = (x, T*S*y).

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