What do mathematicians mean by “equipped”?

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?


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,,), where V is a vector space and ,:V×VC 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 L2[0,1]?), so we refrain from defining these “pairs”, and simply “equip” our space.

Source : Link , Question Author : Olórin , Answer Author : user14717

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