Occupy Hilbert Space is a quirky event proposed by a Facebook friend of mine.   A Hilbert space is a formal mathematical construct in which the information about a physical system is represented by vectors.  In physics we most often encounter the concept of Hilbert spaces in Quantum Mechanics.  They are not limited just to quantum mechanics.  For my submission I will show that we are all occupying a Hilbert space right now.   

Niel Bates describes it like so. 
There's always room for one more and when the police show up they'll never be able to find us.... OHS™ takes place at midnight local time around the globe New Year's Eve 31 DEC 2011. OHS™ is a nonlocal event that takes place outside the boundaries of space and time, whosetheme is for the 99.999...% to protest the hegemony of the Mathematical-Industrial Complex. There is safety in numbers! Please feel free to post your OHS™ protest photos, treatises on multidimensional spaces, and your state vector, or quantum mechanical rants here during the event. Please set your compression ratio to "exponential".
I propose that the familiar 3D space we are in right now is a Hilbert space.  

Stated briefly Hilbert spaces have the following properties.  A vector space H defined over a field of scalars f (most commonly real or complex numbers) which has an inner product is a Hilbert space if. ...

1.) The inner product is unchanged by the conjugate transpose operation.
2.) The inner product is linear in at least it's first argument.   
3.) The inner product is always a positive real number. 

An interesting consequence of these is that any inner product on a Hilbert space defines a distance like function in that space. 

Does the space we are in meet this definition?
The dot product of vectors in 3D Euclidian space is...

As the vectors in Euclidean space are defined over real numbers a conjugate transpose of the elements leaves the result unchanged.

It is also linear in it's first term.

Last but not least it is positive definite. 

In short when you hear a physicist talk about Hilbert space think of a space which resembles Euclidean space. 

What about Minkowski space from Einstein's Relativity?  It almost meets the definition of a Hilbert space but for the last condition.   For Minkowski space the metric is part of the inner product.  This metric makes the space non degenerate, but not positive definite. 

So if every element but the time like one is zero, or just not large enough to cancel the time like element one can have a negative inner product in Minkowski space.   No wonder so many theorist treat time as being somehow different from the other spatial dimensions. 

Tl;Dr  If you want to see an example of a Hilbert space look around you!