Functional Analysis  

• Functional analysis is based on
  • Linear algebra  
    • ---> extended to linear operator in functional analysis
  • Analysis in n-dimensional Euclidean spaces (real analysis, vector analysis, matrix analysis)
    • --->  Hilbert spaces preserve some Euclidean geometric property, e.g., angle and orthogonality by defining inner product (which cannot be done with norm, so Hilbert space is more useful than Banach space in many practical situations); 
    • Parallelogram theorem in Hilbert space has similar flavor as Euclid fifth postulate (existence of unique parallel line)
  • Metric space: define a metric, study convergence, continuity of the mappings between metric spaces
    • --->  study linear operators on metric spaces and continuity of these operators
  • Topological spaces: define topology, open sets, convergence, study continuity of the mappings between topological spaces; Hausdorff property guarantees the uniqueness of convergence of a sequence (like Nested interval principle in Real analysis).
    • --->  study linear operators on topological spaces and continuity of these operators
  • Measure theory including Lebesgue integration: define measures and integrals so that we can study some important function spaces, e.g., L^p, l^p, C^p.
    • --->  study linear operators on measure spaces and continuity of these operators
• Study function spaces from two perspective:
  • Geometric property
    • Metric (metric space) or open set (topological space)
    • Define convergence in terms of either metric or neighborhood (open set)
    • Continuous operators and homeomorphisms (continuous bijective and having continuous inverse) preserve the convergence property from one space to another space.
  • Algebraic property (use abstract algebra: group, ring, field)
    • Define isomorphism, which preserves the algebraic operations from one algebraic system to another algebraic system.
    • Define linear vector space over a field F (e.g., the reals R and the complex C).   
      • A linear vector space is defined with 8 properties (axioms), associative law, distributive law, communitative law, identity element w.r.t. vector addition, inverse element w.r.t. vector addition, etc.
    • Define linear operator, which preserves vector addition and scalar multiplication from one linear vector space to another linear vector space.
• Classification of operators
  • Linear operators, which preserves arithmetic.
  • Continuous operators, which preserves convergence.
  • Bounded operators, which have bounded sup-norm; and Unbounded operators
  • Adjoint operator
  • Compact operator
  • Integral operator
  • Hermitian operator
• Topological space (X, \tau) and metric space (X, d)
  • Topological space is more fundamental than metric space.  
    • Let X be a set and let \tau be a collection of its subsets satisfying
      • Both the empty set and X are in \tau.
      • If C is a subset of \tau, then the union of elements in C is in \tau.
      • If A and B are in \tau, then the intersection of A and B is in \tau.

      Then (X, \tau) is called topological space and \tau is called topology.

    • After defining a topology, a open set is defined as an element of the topology.   (This is brutal force.   Every element of a topology is an open set.  A topology is a collection of open subsets of X.)
    • The definition of topology \tau does not require that a metric d be defined.  If a metric d is defined, then the induced topology \tau is a collection of open sets of X.  
    • Differential topology studies non-metrical differential manifolds (no metric is defined).   Differential geometry studies metrical differential manifolds (a metric is defined).
• Line of thoughts
  • First define topology
  • Then define open sets without defining metric.
  • Define neighborhood of x by any open set containing x.
  • Define closed set
  • Define limit or convergence using open sets/neighborhood.
    • A limit may not unique in a topological space.  To guarantee uniqueness, Hausdorff property, which induces a Hausdorff space.
  • Continuous mapping: Suppose T maps X to Y. T is said to be continuous if for any open set G \in Y, T^{-1}(G) is a open set in X.
  • Nested set theorem: the intersection of a sequence of nested sets has only one element.
  • Isometric
  • Contraction mapping
• L^P is a continuous space (using Lebesgue integral), l^p is a discrete space (using sum)
• Geometric meaning of Cauchy-Schwarz inequality
  • cos \theta <=1,
  • cos \theta = <x,y>/(||x||* ||y||)
• Isometric property, homomorphism, and homeomorphism of two spaces
  • Isometric property characterizes geometric similarity between two spaces.
    • A metric space X is isometric to a metric space Y if there is a bijection f between X and Y that preserves distances.
  • Homomorphism characterizes algebraic similarity between two spaces.
    • A vector space X is homomorphic to a vector space Y if there is a bijection f between X and Y that preserves the arithmetic operations.
  • Homeomorphic mapping
    • Continuous, one-to-one, onto, and having a continuous inverse.
    • If there exists a homeomorphism between two spaces, then the two spaces are homeomorphic.
• Important theorems:
  • The closed graph theorem
  • Banach inverse mapping theorem
  • Hahn-Banach theorems
  • Uniform boundedness principle (Banach-Steinhaus Theorem)
  • The open mapping theorem
  • Baire's Category theorem
• Nice properties about compact sets/spaces
  • A compact metric space (X,d) means a lot.  The following three properties are equivalent
    • compact (having finite basis representation)
    • sequentially compact (every sequence has a convergent subsequence)
    • complete and totally bounded (for each \epsilong>0, there is a finite set {x_i} \in X such that X is contained by the unions of N(x_i,\epsilon).
  • Note 1) there is no need to say a space is closed or open since it's both closed and open.  But compact space means a lot.  2) completeness is only used for a space, not for a subset of a space.   A closed subset is complete as a subspace.
  • For finite-dimensional metric space X, totally boundedness of X is equivalent to boundedness of X.
  • For infinite-dimensional metric space X, totally boundedness is not equivalent to boundedness.  Compactness of an infinite-dimensional metric space means that the infinite-dimensional space can be condensed and represented by a finite cover.  That is why it is called compact.
  • For finite-dimensional metric space, linearity of a mapping implies its continuity.  This is not true for infinite-dimensional spaces.
  • If a mapping is continuous, it maps a compact set onto a compact set. 
  • If a mapping is continuous on a compact set A, it is uniformly continuous on A.  (Nice, isn't it?)
  • If mappings are uniformly continuous, we can interchange limit and infinite summation (or integral) of mappings; interchange infinite summation and differentiation.
  • For a topological space (X, \tau), the topology \tau cannot be too weak so that it is not Hausdorff; \tau cannot be too strong so that it is not compact (no finite basis covering).

Topology vs. sigma-algebra

The definitions of topology and sigma-algebra are different. Topology does not require closedness under complement operation. Note that set theory is applicable in both topology and sigma-algebra. The most important difference is that open sets are defined based on topology, rather than sigma-algebra while measure is defined based on sigma-algebra, rather than topology. With open sets, we study convergence (in topology) and continuity of mappings; with measure, we study integration.