This question is probably a silly one, but I'm not the greatest at coming up with counterexamples. Perhaps someone can lend some insight.

The usual (I think?) definition of a vector space $V$ is a space which is closed with respect to (vector) addition and scalar multiplication (with scalars taken from some field $\mathbb{K}$) and upon which these operations satisfy a number of algebraic axioms (which I won't list here but which are listed [elsewhere](http://en.wikipedia.org/wiki/Vector_space#Definition)). So my question:

Can someone give an example of a pair $(V,\mathbb{K})$ consisting of a space $V$ which is closed with respect to addition and multiplication by elements of **a field** $\mathbb{K}$ but which *isn't* a vector space due to it not satisfying any/all of the vector space axioms? I tried Googling such a thing as well as searching for various related phrases here in SXE but I had no luck.

Any information would be hugely appreciated!

**Edit:** In the third paragraph, I *do* still want $\mathbb{K}$ to be a field. I've added that to the paragraph itself.