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Mathematical Spaces

[last updated: 2024-03-16]
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  • Introduction/Disclaimer:
      I've been curious about Spaces since first learning about them in some analytical class.
      Here are a semi-random collection of notes about the concept.
      Content here is:
        guaranteed to be incomplete,
        likely ambiguous/unclear if not wrong in places,
        but it's a start ...

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  • No Definition:
    • According to wikipedia, mathematics does not define the concept of space itself.
    • In general, however, Spaces consist of a set (also called a universe), with added structure.
      • A Set
        • is a collection of objects
        • The fundamental relationship between an object and a set
          is that of membership.
            If an object a is a member of set A,
            a is said to be an element of A.
            symbolized as:     a ∈ A
        • A set can be defined:
          • by listing its elements, separated by commas,
          • or by defining some property (in braces {} ) that an element must have
            in order to be a member of the set.
              Defining a set by specifying a property must be done carefully,
              because if choice of property is unrestricted,
              it is possible to choose properties that will lead to paradoxes/contradictions,
              such as Russel's paradox, that arises when using the property: the set of all sets not a member of themselves.
        • Set theory is complicated...

        • Set operations (between sets) include union, intersection, and complementation.
      • Structure is defined as ...

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    • Examples of Spaces...
      • Vector Space:
        • A vector space consists of a set of elements called vectors
          There are two operations defined: vector addition, and scalar multiplication
        • There are 8 axioms that must be valid to be considered a "vector space."
          • vector addition is associative.
          • vector addition is commutative.
          • there exists an additive identity (zero vector: v + 0 = v)
          • there exists an additive inverse vector for every vector ( v + (-v) = 0 )
          • scalar multiplication is associative ( a (bV) = (ab)V )
          • there exists a multiplicative identity ( ie: 1 such that: 1v = v )
          • scalar multiplication is distributive across vector addition ( a ( U + V) = aU + aV ).
          • scalar addition is distributive across scalar-vector multiplication ( (a+b) V = aV + bV ).

      • ... other spaces ...

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