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By Tao T.
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By continuity, the inverse image of the unit ball in Y must be an open neighbourhood of 0 in X, thus there exists some radius r > 0 such that T x Y < 1 whenever x X < r. The claim then follows (with C := 1/r) by homogeneity. (Alternatively, one can deduce (iii) from (ii) by contradiction. If (iii) failed, then there exists a sequence xn of non-zero elements of X such that T xn Y / xn X goes to infinity. 3. 16. 2) and is left as an exercise to the reader; the hard part is establishing existence.
By linearity, this implies that λ and λg agree on simple functions. 8) and using continuity (and the finite measure of µ) we conclude that λ and λg agree on all bounded functions. Taking monotone limits (working on the positive and negative supports of the real and imaginary parts of g separately) we conclude that λ and λg agree on all functions in Lp , and in particular that X f g dµ is absolutely convergent for all f ∈ Lp . To finish the theorem in this case, we need to establish that g lies in Lp .
X2n |(x1 , . . 32) (x1 , . . , xn ) · (y1 , . . , yn ) := x1 y1 + . . 33) |x| = (x · x)1/2 . 34) x·x≥0 with equality if and only if x = 0. 36) x · y = y · x. These properties make the inner product easier to manipulate algebraically than the norm. 37) (z1 , . . , zn ) := |z1 |2 + . . 38) (z1 , . . , zn ) · (w1 , . . , wn ) := z1 w1 + . . 39) x = ( x, x )1/2 . 48 1. 40) x, x ≥ 0, with equality if and only if x = 0. 42) x, y = y, x in order to be compatible with sesquilinearity. We can formalise all these properties axiomatically as follows.