Question 5.3.5: Except for the euclidean norm, is any other vector p-norm ge......
Except for the euclidean norm, is any other vector p-norm generated by an inner product?
No, because the parallelogram identity (5.3.7) doesn’t hold when p ≠ 2.
||x + y||^2 + ||x − y||^2 = 2(||x||^2 + ||y||^2) (5.3.7)
To see that ||x + y||^2_p + ||x – y||^2_p = 2(||x||^2_p + ||y||^2_p) is not valid for all x, y ∈ \mathcal{C}^n when p ≠ 2, consider x = e_1 and y = e_2. It’s apparent that ||e_1 + e_2||^2_p = 2^{2/p} = ||e_1 – e_2||^2_p, so
||e_1 + e_2||^2_p + ||e_1 – e_2||^2_p = 2^{(p+2)/p} and 2(||e_1||^2_p + ||e_2||^2_p) = 4.
Clearly, 2^{(p+2)/p} = 4 only when p = 2. Details for the ∞-norm are asked for in Exercise 5.3.7.
Conclusion: For applications that are best analyzed in the context of an innerproduct space (e.g., least squares problems), we are limited to the euclidean norm or else to one of its variation such as the elliptical norm in (5.3.5).
||x||_A = \sqrt{\left\langle x|x\right\rangle} = \sqrt{x^∗A^∗Ax} = ||Ax||_2. (5.3.5)