Question 5.3.2: Describe the norms that are generated by the inner products ......

• Given a nonsingular matrix A ∈ \mathcal{C}^{n×n}, the A-norm (or elliptical norm) generated by the A-inner product on \mathcal{C}^{n×1} is

||x||_A = \sqrt{〈x|x〉} = \sqrt{x^∗A^∗Ax} = ||Ax||_2.                    (5.3.5)

• The standard inner product for matrices generates the Frobenius matrix norm because

||A|| = \sqrt{〈x|x〉} = \sqrt{trace  (A^∗A)} = ||A||_F.                (5.3.6)

• For the space of real-valued continuous functions defined on (a, b), the norm of a function f generated by the inner product 〈f|g〉 = \int_{a}^{b} f(t)g(t)dt is

||f|| = \sqrt{〈f|f〉} = \left(\int_{a}^{b}{f(t)^2 dt}\right)^{1/2}.