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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. 1 (Inner product space). e. e. 40) for all x ∈ V , with equality if and only if x = 0. We will usually abbreviate (V, , ) as V . A real inner product space is defined similarly, but with all references to C replaced by R (and all references to complex conjugation dropped).

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.

Let 0 < p ≤ 1 and f, g ∈ Lp . (i) Establish the variant f + g triangle inequality. p Lp ≤ f p Lp + g p Lp of the (ii) If furthermore f and g are non-negative (almost everywhere), establish also the reverse triangle inequality f + g Lp ≥ f Lp + g Lp . (iii) Show that the best constant C in the quasi-triangle inequal1 ity is 2 p −1 . In particular, the triangle inequality is false for p < 1. 3. Lp spaces 35 (iv) Now suppose instead that 1 < p < ∞ or 0 < p < 1. If f, g ∈ Lp are such that f +g Lp = f Lp + g Lp , show that one of the functions f , g is a non-negative scalar multiple of the other (up to equivalence, of course).