131: MathematicalSpaces

131: MathematicalSpaces

1. Linear Spaces (Vector Spaces)

   • Vector Subspaces
   • contain elements
     • infinite dimensional
       • functions
         • real
           • defining properties
         • complex
     • finite dimensional
       • vectors
         • real
         • complex
           • defining properties
             • closed under vector addition and scalar multiplication (linear combination)
             • Interaction
               • Vector Algebra
                 • Vector Multiplication
                   • Scaling
                   • Two Vectors
                     • Dot Product
                       • Einstein Summation Notation
                       • Algebraic Notation
                       • Intuition
                         • Vector Projection; Observe "overlap degree" of two vector quantities
                         • small-scale operation (vector-on-vector)
                     • Cross Product
                       • Levi-Civita Notation
                         • Levi-Civita (Permutation) Symbol
                           • E123=1
                           • Antisymmetric (swap sign per each index swap)
                       • Algebraic Notation
                       • Intuition
                         • Defines Perpendicular Vector ("area of plane" vector)
                   • Three Vectors
                     • Triple Scalar Product (cyclically permutable)
                       • Defines 3-D Volume (3-D Determinant)
                     • Triple Vector Product (obeys cyclic permutation)
                 • Vector Addition
                   • Component by Component
         • mathematically represent models of multi-element physical systems
       • matrices (nxn or nxm)

2. Topological Spaces

3. Hilbert Spaces

4. others