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I am currently learning about vector spaces and have a slight confusion.

So I know that a vector space is a set of objects that are defined by addition and multiplication by scalar, and also a list of axioms.

I know that a subspace is created from the subset of a vector space and also defined by 3 properties (contain 0 vector, closed addition, closed multiplication by scalar).

Therefore, a vector space is also a subspace of itself. By this definition, every subspace of a vector space is a vector space.

From these definitions, can we say that all vector spaces are also subspaces? Especially since a vector space is a subspace of itself.

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    $\begingroup$ Yes, but note that when we say a vector space is subspace, we always mean it's a subspace of a particular vector space, though sometimes the containing space isn't specified if it's clear from context. $\endgroup$
    – Travis Willse
    Oct 29, 2014 at 15:51

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Yes, any vector space is a subspace of itself.

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Yes, but be careful. If you take one random set and show that elements are closed under substraction, and scalar multiplication (the conditions for subspace) it doesn't mean that it's a vector space. You can say this only if you know that this set is contained in a vector space.

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  • $\begingroup$ How is that? Isn’t every subspace a vector space? $\endgroup$
    – user129393192
    Jun 14 at 2:57

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