5. * Prove 6. Proposition 5.10 (Kernel of a Linear Functional). Let (X,|| ||) be a normed vector space over F. A linear functional f on X is bounded iff kerf is closed. Hint. Prove the "if part" by contrapositive. * Prove Proposition 5.11 (Unboundedness of Hamel Coordinate Functionals). Let (X,|| ||) be an infinite dimensional Banach space over F with a basis B = {x}ier. Then all but a finite number of the Hamel coordinate functionals Xxx= Σ c₁x; c;(x) = c; = F, je I, relative to B are unbounded. iel H Give an example showing that the completeness requirement for the space is es- sential and cannot be dropped. Drove