Suppose A is a closed subalgebra of C_b(K:Y) . It has been shown that if the set of all strong peak function is dense in A, then the Shilov boundary of A exists and it is the closure of ?A, the set of all strong peak points in A. In this thesis, by using variation method we show that the set of all strong peak points is a norming subset of A. As a corollary, we can induce the denseness of strong peak functions on certain spaces. It is shown that, if B x the closed unit ball of the banach space X is the closed convex hull of uniformly strongly exposed points, then for every Banach space Y, the set of norm attaining elements is dense in L(X:Y), the Banach space of all bounded operators from X into Y. we also apply the variation method to investigate the denseness of the set of strongly norm attaining polynomials, when the set of uniformly strongly exposed points of a Banach space X is a norming subset of ?( n X). As a direct corollary, the set of all points at which the norm of ?( n X) isa Frechet differentiable is a G ? dense subset, if the set of all uniformly strongly exposed points of Banach space X is a norming subset of ?( n X). In the last chapter, we will use the Graph theory to get some strongly norm attaining points or complex extreme points. It has been shown that, there exists a one to one correspondence between n dimensional real Cl spaces and certain graphs with n vertices. It gives geometric picture of extreme points of the unit ball of Cl spaces and plays important role to find the strongly norm attaining points of ?( n X). We can find all complex extreme points on a complex Cl space with an absolute norm. The numerical index of a Banach space is a consetant of that space relating the behavior of the numerical radius with that of the usual norm on the Banach algebra of all bounded linear operators on that space. We show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to l n ? .