Let A E L (R", R"), and let the norm | | All be defined (as in class) by ||All = sup { Ah[ }. Show that 14151 (1) | All = sup {|Ahl}; that is, | | All is the supremum of the values |Ah| for h restricted to the unit-sphere in R" (i.e., restricted to the unit-shell in R" ). (2) 11All = sup that is, the supremum of the values | Ah| / h| for h in the punctured-space " \\ {0} (The norm in the numerator, [Ah), is that of R", whereas the norm in the de- nominator, [h|, is that of ". ) Problem 2 Consider the (real-valued) function f : R2 -> R defined by 0 for (r, y) = (0, 0). f(x, y) ry2 for (x, y) # (0, 0). (a) Prove that f is bounded in 2 (b) Prove that f is not continuous at the origin (0, 0). (Hint: Using limits might prove useful here!) (c) Find the partial derivatives of f at the origin; that is, find - (0, 0) and of (0, 0). (Hint: The limit-definition of partial derivatives might prove useful here!) [Note: Observe that f is not continuous at the origin (0, 0), and therefore is not Frechet(!) differentiable there. Yet, it has partial derivatives there!]