Problem 2: Metric Spaces of Continuous Functions and Graph Representations Background: Consider the space C([0, 1], R) of continuous real-valued functions on the interval [0, 1], equipped with the supremum norm ||f||∞ = sup Tasks: Pze[0,1] ||f(x)|. a. Graph Construction for Function Spaces: Define a graph H where each vertex represents a function f = C([0, 1], R). Connect two vertices ƒ and g with an edge if ||fg||∞ < € for some fixed € > 0. Discuss the resulting graph's properties, such as connectivity and diameter, in relation to the completeness and compactness of C([0, 1], R). b. Isometry and Graph Isomorphism: Suppose : C([0, 1], R) → C([0, 1], R) is an isometry with respect to the supremum norm. Show that induces an isomorphism between the corresponding graphs H. Use this to deduce properties about &, potentially leveraging the Banach-Stone theorem. c. Graph-Based Compactness Criteria: Utilize the graph H to formulate and prove a compactness criterion for subsets of C([0, 1], R). Specifically, relate the Arzelà-Ascoli theorem to graph-theoretic notions such as finite coverings or graph limits. d. Embedding Metric Spaces into Graphs: Given a general metric space (X, d), describe a procedure to embed X into a graph in a way that preserves metric properties relevant to functional analysis (e.g., completeness, separability). Apply this procedure to C([0, 1], R) and discuss the implications for functional analysis on this space. 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: (M1) d is real-valued, finite and nonnegative. (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). ■