Though this may be related to lin. alg. but it deals with Analysis.(adsbygoogle = window.adsbygoogle || []).push({});

There are 8 axioms for Vector Spaces. To prove a space ##V## is a VS, one must check all 8 axioms (i.e. closure under addition, scalar multi. etc...)

My question is this, it seems cumbersome to have to do this every time. Would it be better to use the lemma that states: "A non∅ subset ##W## or a VS ##V## is a subspace ##iff##

##\alpha v + \beta w \in W, \forall \ v, w \in W, \ \alpha , \beta \in \mathbb{F} \ \ (1)##where ##\mathbb{F}## is the field of scalars.

First, would it be correct to use this "Lemma"?

And second, what should the encompassing VS be: because the lemma states ##W## is a subspace of a VS ##V##, but if I want to prove a set

##G \subsetneq \mathbb{R \times R}## is a VS, I guess I should aim to show it is subspace of the Vector Space ##\mathbb{R \times R}##

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# Proving V is a VS

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