Functional Analysis

Functional Analysis is a branch of mathematical analysis that extends the methods of linear algebra and calculus from finite-dimensional vector spaces to infinite-dimensional spaces whose elements are functions. It treats entire functions as single points and studies these function spaces (such as Banach spaces and Hilbert spaces) by equipping them with a notion of size or distance, known as a norm. This framework allows for the rigorous study of concepts like convergence, continuity, and transformations (operators) in an infinite-dimensional setting, providing the essential mathematical language for theories like quantum mechanics, partial differential equations, and signal processing.

1. Preliminaries and Foundational Concepts
   1. Review of Linear Algebra
      1. Vector Spaces
         1. Definition and Axioms
         2. Examples of Vector Spaces
         3. Subspaces
            1. Definition and Characterization
            2. Examples of Subspaces
            3. Intersection and Sum of Subspaces
         4. Linear Combinations
         5. Linear Span
            1. Spanning Sets
            2. Properties of Span
      2. Linear Independence and Dependence
         1. Definition of Linear Independence
         2. Criteria for Linear Independence
         3. Linear Dependence Relations
         4. Maximal Linearly Independent Sets
      3. Basis and Dimension
         1. Definition of Basis
         2. Existence of Bases
         3. Uniqueness of Dimension
         4. Finite and Infinite Dimensional Spaces
         5. Coordinate Representation
      4. Linear Transformations
         1. Definition and Basic Properties
         2. Examples of Linear Transformations
         3. Kernel and Image
            1. Properties of Kernel
            2. Properties of Image
         4. Rank-Nullity Theorem
         5. Invertible Linear Transformations
      5. Matrix Representations
         1. Matrix of a Linear Transformation
         2. Change of Basis
         3. Similarity of Matrices
   2. Metric Spaces
      1. Definition and Basic Properties
         1. Definition of a Metric
         2. Metric Axioms
         3. Examples of Metrics
            1. Euclidean Metric
            2. Discrete Metric
            3. Taxicab Metric
            4. Maximum Metric
      2. Topology of Metric Spaces
         1. Open Balls and Neighborhoods
         2. Open Sets
            1. Properties of Open Sets
            2. Basis for Topology
         3. Closed Sets
            1. Characterization of Closed Sets
            2. Closure of a Set
         4. Interior and Boundary
      3. Sequences and Convergence
         1. Convergent Sequences
         2. Uniqueness of Limits
         3. Subsequences
         4. Cauchy Sequences
      4. Completeness
         1. Complete Metric Spaces
         2. Examples of Complete Spaces
         3. Examples of Incomplete Spaces
         4. Completion of Metric Spaces
      5. Continuous Functions
         1. Definition of Continuity
         2. Sequential Characterization
         3. Uniform Continuity
         4. Lipschitz Continuity
         5. Homeomorphisms
      6. Compactness
         1. Definition via Open Covers
         2. Sequential Compactness
         3. Equivalence in Metric Spaces
         4. Heine-Borel Theorem
         5. Properties of Compact Sets
      7. Connectedness
         1. Connected Sets
         2. Path Connectedness
         3. Components
      8. Important Theorems
         1. Cantor's Intersection Theorem
         2. Baire Category Theorem
         3. Contraction Mapping Theorem
   3. Topological Spaces
      1. Basic Definitions
         1. Topology on a Set
         2. Open and Closed Sets
         3. Examples of Topological Spaces
      2. Topological Axioms
         1. Axioms for Open Sets
         2. Axioms for Closed Sets
      3. Bases and Subbases
         1. Basis for a Topology
         2. Generating Topology from Basis
         3. Subbases
         4. Neighborhood Bases
      4. Continuous Functions
         1. Definition in Topological Setting
         2. Characterizations of Continuity
         3. Homeomorphisms
      5. Separation Axioms
         1. T₀ Spaces
         2. T₁ Spaces
         3. T₂ (Hausdorff) Spaces
         4. Regular and Normal Spaces
      6. Compactness in Topological Spaces
         1. Compact Spaces
         2. Locally Compact Spaces
         3. Compactification
      7. Connectedness in Topological Spaces
         1. Connected Spaces
         2. Path Connected Spaces
         3. Locally Connected Spaces