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The crazy vector space is a vector space.

Created over 8 years ago, updated 10 days ago

The "crazy" vector space is an example that shows the operations of addition and scalar multiplication need not look "normal" to satisfy the vector space axioms.

Example: Let (V = \mathbb{R}^+) (positive real numbers) with:

  • "Addition": (x \oplus y = xy) (ordinary multiplication)
  • "Scalar multiplication": (c \odot x = x^c)

Verification:

  • Zero vector: (1) (since (x \oplus 1 = x \cdot 1 = x))
  • Additive inverse of (x): (1/x) (since (x \oplus (1/x) = x \cdot (1/x) = 1))
  • (c \odot (x \oplus y) = (xy)^c = x^c y^c = (c \odot x) \oplus (c \odot y))

This space is isomorphic to (\mathbb{R}) via the logarithm map: (\log(x \oplus y) = \log(xy) = \log x + \log y).

The purpose of this example is to emphasize that vector spaces are defined by their axioms, not by the appearance of their operations.