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). ■

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). ■

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter5: Polynomial And Rational Functions

Section5.3: Graphs Of Polynomial Functions

Problem 2TI: Use the graph of the function of degree 5 in Figure 10 to identify the zeros of the function and...

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