I have this exercise that asks me to prove the set V=R is a vector space.
Operations are
$ x \oplus y = \sqrt[3]{x^3+y^3} $
and
$ c \otimes x = \sqrt[3]{c*x} $
I already proved all axioms until Distributive I'm getting a bit confused here, do i have to consider $ \otimes $ and $ \oplus$ as operations always ? (except for under the root)
For example, can i do this ?
$ c \otimes (x + y) = \\ \sqrt[3]{c*(x+y)} = \\ \sqrt[3]{c*x + c*y} $
As you see i used $\otimes$ and normal +
Or is this the right way ?
$ (c \otimes x) \oplus (c \otimes y) = \\ \sqrt[3]{c*x} \oplus \sqrt[3]{c*y} = \\ \sqrt[3]{(\sqrt[3]{c*x})^3 + (\sqrt[3]{c*y})^3} = \\ \sqrt[3]{c*x + c*y} $
AND
$ c \otimes (x \oplus y) =\\ \sqrt[3]{c*(x\oplus y)} = \\ \sqrt[3]{c*x \oplus c*y} \\ \sqrt[3]{\sqrt[3]{(c*x)^3 + (c*y)^3}}$
In this case distribution seems not to work as $\sqrt[3]{\sqrt[3]{(cx)^3 + (cy)^3}}$ is not equal to $\sqrt[3]{cx + cy}$
Can someone help me do distribution the right way ? Thank you
You have copied the definition wrongly. The correct definition of $c \otimes x$ is $c \otimes x=(\sqrt [3] c)x$.