By G. H. Hardy

There will be few textbooks of arithmetic as famous as Hardy's natural arithmetic. considering its e-book in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have grew to become initially in their undergraduate classes. In its pages, Hardy combines the passion of a missionary with the rigor of a purist in his exposition of the elemental rules of the differential and fundamental calculus, of the homes of limitless sequence and of alternative subject matters related to the proposal of restrict.

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82) ˜k (x) = uk (x)/ρk → 0, and, consequently, u ˜(x) = 0. 82) holds as well. If x ∈ Ω3 , there are subsequences for which |uk (x)| → ∞ and subsequences for which |uk (x)| is bounded. 82) holds, and for the latter it holds as well since u˜(x) = 0. e. on the whole of I. e. 78). 18) F (x, uk )/ρ2k dx → 2 I β(x)˜ u(x)2 dx. 84) I Hence, 2G(uk )/ρ2k → 1 − β(x)˜ u(x)2 dx I = (1 − u ˜ 2 H) + u ˜ 2 [1 − β(x)]˜ u(x)2 dx + I = A + B + C. Since u ˜ H ≤ 1 and β(x) ≤ 1, the quantities A, B, C are each ≥ 0. The only way the sum can equal 0, is if each equals 0.

2 |k|≤n Consequently, |αk |2 ≤ u 2 . |k|≤n Since this is true for every n, we have ∞ |αk |2 ≤ u 2 . 54) k=−∞ Moreover, if m < n, then un − um 2 |αk |2 → 0 = m<|k|≤n as n → ∞. 4 Fourier series 19 Hence, {un } is a Cauchy sequence in L2 (I). It converges in L2 (I) to a function u ˜. Let k = 0, ±1, ±2, . . u, ϕ¯k ), α ˜ k = (˜ Then, α ˜ k = ( lim un , ϕ¯k ) = lim ( n→∞ n→∞ αj ϕj , ϕ¯k ) = lim n→∞ |j|≤n αj (ϕj , ϕ¯k ) = αk . |j|≤n Let f = u ˜ − u. 51). 19 that f ≡ 0. 45) holds. 17. Proof. Assume that there is a point x0 ∈ I such that f (x0 ) > 0.

Let M be the subspace of those functions in H which are orthogonal to N, that is, functions w ∈ H which satisfy (w, 1)H = w(x) dx = 0. 23. 72) for some constants a, b. We shall also prove these at the end of the section. 24. 67) there is a u in H such that G(u) = min G. 2) in the usual sense. Proof. Let α = inf G, H and let {uk } be a minimizing sequence, that is, a sequence satisfying G(uk ) α. Assume ﬁrst that ρk = uk H ≤ C. 73) and uk (x) → u0 (x) uniformly in I. 74) Then F (x, uk )dx → I F (x, u0 )dx I by arguments given previously.