Answer:

Definition of Vector Space:
A vector space is a set of vectors which satisfy some axioms such as closure under addition and scalar multiplication. The axioms include the existence of an additive identity element, zero, and additive inverse element, and distributivity of scalar multiplication over vector addition.

The Vector Space with Only the Additive Identity Element:
The vector space with only the additive identity element is also known as the trivial vector space. It is a vector space that contains only the zero vector, which is the additive identity element. The trivial vector space is denoted by {0}.

Option 1: Real Space
The set of all real numbers is a vector space. However, it is not a trivial vector space because it contains more than one vector. It includes the zero vector, but it also contains all non-zero real numbers, which are closed under addition and scalar multiplication.

Option 2: Complex Space
The set of all complex numbers is also a vector space. However, it is not a trivial vector space because it contains more than one vector. It includes the zero vector, but it also contains all non-zero complex numbers, which are closed under addition and scalar multiplication.

Option 3: Null Space
The null space, also known as the kernel, is a vector space that contains only the zero vector. It is a subspace of any vector space, and it is always a trivial vector space.

Option 4: Rank Space
The rank space, also known as the range, is the set of all possible outputs of a linear transformation. It can be a non-trivial vector space, depending on the linear transformation.

Conclusion:
The only option that is a trivial vector space is the null space. Therefore, the correct answer is Option 3: Null space.