TITLE: What do mathematicians mean by "equipped"?
QUESTION [51 upvotes]: I am a mathematical illiterate, so I do not know what people mean when they say "equipped".
For example, I say that a Hilbert space is a vector space equipped with an inner product. What does that actually mean? 
Obviously, one interpretation is to picture professor Hilbert as a plumber with an extra tool hanging out of his back pocket (a.k.a. an inner product), but mathematically why can't we do the inner product in a vector space? 
Both Hilbert space and vector space work with functions and vectors, don't they? 
Why can't we define a space where all operations are possible?

REPLY [7 votes]: Fundamentally, "equipped with" just means "and". A Hilbert space is an object which consists of two components: a vector space, and an inner product on that vector space.
This means that strictly speaking, a Hilbert space and a vector space are different objects: a Hilbert space is not a vector space, and a vector space is not a Hilbert space.
But the reason that a mathematician writes "equipped with" is that they don't want to speak strictly; they want to speak loosely, because that's easier. So what your sentence means is this:

A Hilbert space is an object which consists of two components: a vector space, and an inner product on that vector space. However, I expect it to be convenient to speak as if a given Hilbert space were the same thing as the vector space which is its main component, and so I will do so.

And so, thenceforth, whenever the mathematician talks about a Hilbert space $V$, they will often speak as if $V$ were a vector space, rather than merely having a vector space as a component.
As for why you can't do inner products in vector spaces in general? Well, nobody has ever defined the word "inner product" in a way which makes sense for every vector space. You can't take an inner product until you've decided what the phrase "inner product" means in your specific case.