91Ó°ÊÓ

In vector spaces, one essential property is the closure under addition. This means that if you take two elements (vectors) from the vector space and add them, the result should still be a part of that same space. In mathematical terms, if \(x\) and \(y\) are elements of the vector space, then \(x \oplus y\) should also belong to the vector space.
To relate this to our problem, the operation of addition is defined as \(x \oplus y = xy\) for the set of all positive real numbers. Since multiplying two positive real numbers results in a positive real number, this set is closed under the defined operation.
However, closure under addition is just one of the many requirements a set must fulfill to be a vector space. Without meeting other necessary conditions, the set can't be considered a vector space. But, for closure under addition, our given set does indeed meet this requirement.

A key component of a vector space is the existence of an additive identity. The additive identity is an element within the vector space which, when added to any element of the vector space, returns the original element unchanged. It's like having a neutral number that doesn't affect the original value when combined.
For traditional numbers, the additive identity is zero, since \(x + 0 = 0 + x = x\). In our exercise, addition is defined differently, as \(x \oplus e = xe\), meaning it needs an element \(e\) such that \(x \oplus e = x\) for any \(x\) in the space.
Looking at our configuration, there doesn't exist a positive real number \(e\) that satisfies \(xe = x\) for all positive real \(x\). Therefore, without an additive identity element, this set fails to be a vector space.

Distributive properties are fundamental to the structure of a vector space. They are split into two separate axioms, involving both scalar multiplication and addition.
The first distributive property requires that a scalar multiplied by the addition of two vectors should yield the same result as multiplying the scalar with each vector individually and then adding the two outcomes. Specifically, the failure here is seen when we want \(c \odot (x \oplus y)\) to equal \((c \odot x) \oplus (c \odot y)\). With our defined operations, this translates into \((xy)^c\) not being the same as \(x^c y^c\), unless under very special conditions, like when both \(x\) and \(y\) are the same.
The second distributive property deals with two scalars being added and then used for scalar multiplication of a vector, which should equal individual scalar multiplications added together: \((c + d) \odot x\) should be \(c \odot x \oplus d \odot x\). The mathematical breakdown shows it as \((c+d)x = x^c x^d\), but this isn't generally true in our setting.
Because the defined operations do not fulfill these distributive properties, the given set cannot function as a vector space, as it cannot satisfy all necessary vector space axioms.