Functional Analysis HW # 5

نویسندگان

  • HW
  • Sangchul Lee
چکیده

Proof. In view of [Dou98, Proposition 3.11], it suffices to check that the supremum norm invalidates the parallelogram law. To this end, let us choose a very specific function f (x) = cos2(2πx) and g(x) = sin2(2πx). Then we have f (x) + g(x) = 1 and f (x) − g(x) = cos(4πx), hence ‖ f + g‖ = ‖ f − g‖ = 1. On the other hand, we have ‖ f ‖ = ‖g‖ = 1. Consequently ‖ f + g‖ + ‖ f − g‖ = 2 , 4 = 2‖ f ‖ + 2‖g‖. Therefore the supremum cannot be induced by an inner product.

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تاریخ انتشار 2015