By Roger Godement
Research quantity IV introduces the reader to useful research (integration, Hilbert areas, harmonic research in workforce idea) and to the tools of the speculation of modular services (theta and L sequence, elliptic features, use of the Lie algebra of SL2). As in volumes I to III, the inimitable variety of the writer is recognizable right here too, not just due to his refusal to write down within the compact kind used these days in lots of textbooks. the 1st half (Integration), a smart mixture of arithmetic stated to be 'modern' and 'classical', is universally necessary while the second one half leads the reader in the direction of a truly lively and really expert box of study, with probably extensive generalizations.
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Extra info for Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights (Universitext)
The same argument applies to functions |gn |, which converge ae. to |f |p , whence µ (|f |p ) = lim µ (|gn |) = lim µ (|sn |p ) = lim Np (sn )p since the sn ∈ L(X) are integrable. Since the series Lp , it finally follows that fn converges to f in µ (|f |p ) = Np (f )p , whence (2). Conversely, suppose that g = |f |p−1 f ∈ Lp . The theorem being trivial for p = 1, one may suppose that p > 1. The map z → |z|p−1 z from C to C is a homeomorphism. Its inverse is z → |z|−1/q z, with 1/p + 1/q = 1 and so 0 < 1/q < 1 .
Let us suppose that f ∈ Lp , choose (corollary 1 of theorem 7) a series of functions fn ∈ L(X) such that Np (fn ) < +∞ , f (x) = fn (x) ae. and set sn (x) = f1 (x) + . . + fn (x) , S(x) = |fn (x)| . § 2. Lp Spaces 31 The functions gn = |sn |p−1 sn are still in L(X) since p > 1. On the other hand, |gn | = |sn |p ≤ S p = G with a function G having values in [0, +∞] satisfying p N1 (G) = N1 (S p ) = Np |fk | p ≤ Np (fk ) < +∞ . The dominated convergence theorem can therefore be applied in L1 to the sequence (gn ), and as it converges ae.
This result explains its importance in applications since the theory of Hilbert spaces is far simpler and more complete than that of other classes of topological vector spaces. We will again come across the L2 spaces in relation to Fourier transforms, but they have many other applications. 14 One can formulate the same definition on F 2 , but the result would not be an inner product: the function µ∗ is not linear on F 1 . § 2. 9) |µ(f g)| ≤ N1 (f g) ≤ Np (f )Nq (g) . As a result, for given g ∈ Lq , the map f −→ µ(f g) is a continuous linear functional on Lp .