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Often times in introductory linear Algebra the statement Prove that $V$ is vector space comes up.

My question is whether proving closure under addition and scalar multiplication along with the existence of an additive identity is sufficient to show that some set is a vector space or do we need to prove the other plethora of properties that go along with the definition of a vector space.

I ask this because generally those other properties can be inferred from the basic properties of closure under addition and scalar multiplication.

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A minimal set of axioms for vector spaces is:

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This is discussed and proved in this paper:

Independent Axioms for Vector Spaces, J. F. Rigby and James Wiegold. The Mathematical Gazette Vol. 57, No. 399 (Feb., 1973), pp. 56-62.

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If you're given that $V$ is a vector space and $U$ some space in $V$ then to show that $U$ is a subspace you only have to show that $U$ is closed under addition and scalar multiplication. However, more generally, if you're trying to show that some space $V$ is a vector space you have to show that it satisfies all the axioms of a vector space.

Additionally, closure under multiplication and addition does not guarantee that the vector axioms are satisfied.

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