# 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 Oct 29 '14 at 15:51

## 2 Answers

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.

Yes, any vector space is a subspace of itself.