Mathematical Terminology – What Does ‘Equipped’ Mean?

Question

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?

Answer 1

The word "equipped" keeps notational pandemonium from breaking loose. For instance, if you were to be a bit more formal, you'd say

A Hilbert space is a pair $(V, \left<\cdot,\cdot \right>)$, where $V$ is a vector space and $\left<\cdot,\cdot \right>\colon V\times V \to \mathbb{C}$ is an inner product. Additionally, all Cauchy sequences in $V$ are convergent in the norm induced by the inner product to an element in $V$.

But most of the time, there's no reason to disambiguate between the vector space and the inner product (who puts a different inner product on the set $L^2[0,1]$?), so we refrain from defining these "pairs", and simply "equip" our space.