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). ■ Problem 1: Compact Operators on Hilbert Spaces and Graphs Background: Let H be a separable Hilbert space, and consider the space of bounded linear operators B(H) endowed with the operator norm. Compact operators form a closed subset K(H) CB(H). Tasks: a. Graph Construction: Construct a graph G whose vertices correspond to an orthonormal basis {en} of H. Define edges based on the action of a compact operator T = K(H) such that there is an edge between e, and em if (Ten, em) 0. Describe the properties of G that reflect the compactness of T. b. Spectral Graph Analysis: Suppose T is a compact, self-adjoint operator. Using the graph G from part (a), analyze how the spectral properties of T (eigenvalues and eigenvectors) are represented within the graph structure. Provide a detailed correspondence. c. Graph Limits and Operator Norms: Investigate how sequences of such graphs {G} corresponding to a sequence of compact operators {T} can converge in a suitable graph metric (e.g., Gromov- Hausdorff distance). Relate this convergence to the convergence of T in the operator norm on K(H). d. Application to Functional Analysis: Prove that the space K(H) is complete with respect to the operator norm by utilizing the graph constructions and properties established in parts (a)-(c).