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))
**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.