Solving gram-schmidt orrthonormalization

Question

Problem:

Gram-Schmidt Orrthonormalization

1. Show that the norm ||A|| = max_i,j |a_ij| on the space of n×n real matrices is not induced by any norm in ℜⁿ . Hint: use 8.10(ii).

2. Prove the Neuman lemma: if ||A|| < 1, then I - A is invertible. Here || · || is a norm on the space of n × n matrices induced by a norm ℜⁿ or ?ⁿ

3. Let V be an inner product space, and || · || be the norm induced by the inner product.

Prove the parallelogram law

                                 ||u + v||² + ||u - v||² = 2||u||² + 2||v||²

Based on this, show that the norms || · ||₁ and || · ||_∞ in ℜ² are not induced by any inner product.

4. Let {w₁, . . . , w_n} be an ONB in ℜⁿ . Assuming that n is even, compute
                            ||w₁ - w₂ + w₃ - · · · - w_n||

5. Apply Gram-Schmidt orthonormalization to the the basis {1, x, x²} in the space P₂(ℜ) with the inner product

                                                {f,g} = ∫¹₀ ƒ(x)g(x)dx

Extra credit: Find two norms on the space C[0, 1] that are not equivalent.