# Are all vector spaces also a subspace?

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.

• 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. – Travis Willse Oct 29 '14 at 15:51