／var／logmarcus chiu

Mathematical Spaces

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Mathematical Spaces
  • is, informally, a collection/set/universe of mathematical objects that are treated as “points” with some selected structure/relationship(s) between these points
  • it is the relationships that define the nature of the space

Mathematical Spaces - Types of Relationships/Structure Between Points

Relationship/Structure Type

What Does it Measure?

Description

closeness

distances

lengths

angles

Topology (𝜏)

✅

❌

❌

❌

Distance Metric (𝑑)

✅

✅

❌

❌
  • a distance metric (𝑑) induces a topology (𝜏𝑑) where the set { 𝐵𝑑(𝑥,𝜖) | 𝑥∈𝑋 and 𝜖>0 } forms the basis of a metric topology (𝜏𝑑), where 𝐵𝑑(𝑥,𝜖) is anepsilon balldefined as𝐵𝑑(𝑥,𝜖) = {𝑦∈𝑋 :𝑑(𝑥,𝑦) <𝜖 }

Norm (||·||)

✅

✅

✅

❌
  • a norm (||·||)induces a distance metric (𝑑||·||) defined as: 𝑑||·||(𝑥,𝑦) = ||𝑥-𝑦||

Inner Product (⟨·,·⟩)

✅

✅

✅

✅
  • an inner product (⟨·,·⟩) induces a norm (||·||⟨·,·⟩) defined as: ||·||⟨·,·⟩ = √⟨·,·⟩

Mathematical Spaces - Other

Mathematical Spaces - Transformations From 1 Space to Another

see Morphisms

Mathematical Spaces - Types

Mathematical Spaces - Diagram

An arrow from space 𝐴 to space 𝐵 implies that space 𝐴 is also a kind of space 𝐵 (e.g. a normed vector space is also a metric space)

------ Euclidean Vector Space (ℝᴺ) ----------------
    |                 |                               |

    |                 |                               |

    |                 v                               v

    |           Banach Spaces <---------------- Hilbert Spaces
    |       (norm and completeness)    (inner-product and completeness)

    |                 |                               |

    |                 |                               |

    |                 v                               v

    |         Normed Vector Space <---------- Inner Product Spaces
    |               (norm)                      (inner product)

    |                 |\_____________________________ |

    |                 |                              \|

    |                 v                               v

    |           Metric Spaces                 Locally Convex Space
    |         (distance metric)                  (semi-norm)

    |                 | _____________________________/|

    |                 |/                              |

    v                 v                               v

Manifold ---> Topological Spaces                  Vector Space
            (topology and open set)           (linear combination)

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