Necessary and Sufficient condition for W of vector space to be a subspace is αX + βy belongs to W

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A non-empty subset W of vector space is subspace of V iff αx+βy ∈ W for all α,β∈F and x,y∈W A non-empty subset W of vector space is subspace of V iff αx+βy ∈ W for all α,β∈F and x,y∈W
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A non-empty subset W of a vector space V(F) is a subspace of V if and only if x+y ∈ W and α.x ∈ W. A non-empty subset W of a vector space V(F) is a subspace of V if and only if x+y ∈ W and α.x ∈ W.
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The Necessary and Sufficient Condition theorem on Vector Spaces. The Necessary and Sufficient Condition theorem on Vector Spaces.