Problem 3: Banach Spaces, Duality, and Graph-Theoretic Representations Background: Let X be a Banach space, and X* its dual space. Consider the weak* topology on X*, where convergence is defined pointwise on X. Tasks: a. Graph Construction for Dual Spaces: Construct a graph D where each vertex represents a functional ƒ € X*. Connect two vertices ƒ and g with an edge if |f(x) - g(x)| < 6 for all x in a finite, dense subset {1, 2,..., n} CX and for some 6 > 0. Analyze how the topology of D reflects the weak* topology on X*. b. Graph-Theoretic Duality: Explore how duality between X and X* can be interpreted through graph constructions. For instance, investigate whether properties like reflexivity or the Radon- Nikodym property can be characterized via specific graph-theoretic features of D. c. Weak* Compactness and Graph Properties: Use the graph D to establish a characterization of weak* compact sets in X*. Specifically, relate notions such as total boundedness or closure in the graph metric to classical compactness criteria in functional analysis. d. Operator Graphs and Duality Mappings: Given a bounded linear operator T: XY between Banach spaces, construct a corresponding graph that represents the adjoint operator T* : Y* → X*. Analyze how properties of T, such as being compact or isometric, are reflected in the structure of this graph. 3 K 5 8 K -4.2 -2.5 0 1.7 d(1.7, 2.5) 11.7 (-2.5) |= 4.2 - = - d(3, 8) 13 81-5 Fig. 2. Distance on R x, y = R. Figure 2 illustrates the notation. In the plane and in “ordi- nary" three-dimensional space the situation is similar. In functional analysis we shall study more general "spaces" and "functions" defined on them. We arrive at a sufficiently general and flexible concept of a "space" as follows. We replace the set of real numbers underlying R by an abstract set X (set of elements whose nature is left unspecified) and introduce on X a "distance function" which has only a few of the most fundamental properties of the distance function on R. But what do we mean by "most fundamental"? This question is far from being trivial. In fact, the choice and formula- tion of axioms in a definition always needs experience, familiarity with practical problems and a clear idea of the goal to be reached. In the present case, a development of over sixty years has led to the following concept which is basic and very useful in functional analysis and its applications. 1.1-1 Definition (Metric space, metric). A metric space is a pair (X, d), where X is a set and d is a metric on X (or distance function on X), that is, a function defined² on XXX such that for all x, y, z= X we have: d is real-valued, finite and nonnegative. (M1) (M2) d(x, y)=0 if and only if x = y. (M3) d(x, y) = d(y, x) (Symmetry). (M4) d(x, y)d(x, z)+d(z, y) (Triangle inequality). ■

Use the graph of the function of degree 5 in Figure 10 to identify the zeros of the function and their multiplicities.

(a) Identifying whether a function is a convex function is useful because a convex function's local minimum has the nice property that it has to be the global minimum. Please select all functions below that are convex functions. Note dom(f ) denotes the domain of the function f . Select all that apply: Of (x) =x3 + 3x + 7, domf ) =R Of (x) Of (x) =|x| , domf ) =R Of (x) =2-X, domf) =R Of =||x|, domf) =R" log x, domf) =R,. (the set of positive real numbers) O None of the above. (b) Typically we can solve linear regression problems in two ways. One is through direct methods, e.g. solving the closed form solution, and the other is through iterative methods (gradi- ent descent). Consider a linear regression on data (X, y). We assume each row in X denotes one input in the dataset. Please select all correct options. Select all that apply: O If the matrix X' x is invertible, the exact solution is always preferred for solving the solution to linear regression as computing matrix inversions and…

Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) * Give appropriate graphs and required codes. * Make use of inequalities if you think that required. *You are supposed to use kreszig for reference. (1.2) Definition: A space X is said to satisfy the T-axiom or is said to be a T-space if for every two distinct points x and y = X, there exists an open set containing x but not y (and hence also another open set contain- ing y but not x). Again, all metric spaces are 7₁. It is obvious that every T₁ space is also To and the space (R, I) above shows that the converse is false. Thus the T₁-axiom is strictly stronger than To. (Sometimes a beginner fails to see any difference between the two conditions. The essential point is that given two distinct points, the To-axiom merely requires that at least one of them can be separated from the other by an open set whereas the T-axiom re- quires that each one of them can be separated from the other.) The following proposition…