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The crazy vector space is a vector space.

Created over 8 years ago, updated about 1 month 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.