The solutions to Kreyszig Chapter 3 demonstrate that the geometric properties of Euclidean space (like the Pythagorean theorem and Parallelogram law) extend to all , and that completeness is the defining feature that upgrades an inner product space to a Hilbert space .
Forgetting that in complex spaces, the inner product is conjugate linear in the second argument. Kreyszig’s exercises carefully mix real and complex cases. kreyszig functional analysis solutions chapter 3
Thus it is an inner product (a weighted Euclidean inner product). The induced norm is (|x| = \sqrt\sum w_k x_k^2). The solutions to Kreyszig Chapter 3 demonstrate that