The structure and geometry of subspaces of a given Banach space is among the most fundamental questions in Functional Analysis. In 1961, Kadec and Pelczyński pioneered a field of study by analyzing the structures of subspaces and basic sequences in L_p[0,1] under a naturally occurring restriction of p, 2 < p <\infty. They proved that any infinite-dimensional subspace X\subset L_p[0,1] for 2<p<\infty must either be isomorphic to l_2 and complemented in L_p or must contain a complemented subspace which is isomorphic to l_p. Many works since have studied the relationships between the sides of this dichotomy, chiefly by weakening hypotheses on side of the equation to gain stronger assumptions on the other. In this way, Johnson and Odell were able to show in 1974 that if X contains no further subspace which is isomorphic to l_2, then it must embed into l_p. Kalton and Werner further strengthened this result in 1993 by showing that such an embedding must be almost isometric. We start by analyzing the tools and definitions originally introduced in 1961 and define a natural extension to these methods. By analyzing this extension, we provide a constructive and streamlined reproving of Kalton and Werner's theorem: Let X be …

In a stopping rule problem, a real-time player decides to stop or continue at stage n based on the observations up to that stage, but in a k-step look-ahead stopping rule problem, we suppose the player knows k steps ahead. The aim of this Ph.D. dissertation is to study this type of prophet problems for simple random walk, determine the optimal stopping rule and calculate the expected return for them. The optimal one-step look-ahead stopping rule for a finite simple random walk is determined in this work. We also study two infinite horizon stopping rule problems, sum with negative drift problems and discounted sum problems. The optimal one, two and three-step look-ahead stopping rules are introduced for the sum with negative drift problem for simple random walk. We also compare the maximum expected returns and calculate the upper bound for the advantage of the prophet over the decision maker. The last chapter of this dissertation concentrates on the discounted sum problem for simple random walk. Optimal one-step look-ahead stopping rule is defined and lastly we compare the optimal expected return for one-step look-ahead prophet with a real-time decision maker.