I hope it will HELP youStep-by-step explanation:1. The Hilbert space L2 157 The resulting L 2 (R d )-norm of f is defined by kfkL2(Rd) = µZ Rd |f(x)| 2 DX¶1/2 . The reader should compare those definitions with these for the space L 1 (R d ) of integrable functions and its norm that were described in Sec- tion 2, Chapter 2. A crucial difference is that L 2 has an inner product, which L 1 does not. Some relative inclusion relations between those spaces are taken up in Exercise 5. The space L 2 (R d ) is naturally equipped with the following inner prod- uct: (f, g) = Z Rd f(x)g(x) dx, whenever f, g ∈ L 2 (R d ), which is intimately related to the L 2 -norm since (f, f) 1/2 = kfkL2(Rd) . As in the case of integrable functions, the condition kfkL2(Rd) = 0 only implies f(x) = 0 almost everywhere. Therefore, we in fact identify func- tions that are equal almost everywhere, and define L 2 (R d ) as the space of equivalence CLASSES under this identification. However, in practice it is often convenient to think of elements in L 2 (R d ) as functions, and not as equivalence classes of functions. For the definition of the inner product (f, g) to be meaningful we need to know that fg is integrable on R d whenever f and g belong to L 2 (R d ). This and other basic properties of the space of square integrable functions are gathered in the next proposition. In the rest of this chapter we shall denote the L 2 -norm by K · k (drop- ping the subscript L 2 (R d )) unless stated otherwise