I have read about Hilbert spaces, and I have forgotten what I read. What does
it mean if I nonetheless have, or think I have some understanding, or at least
some "intuition", of what Hilbert spaces might be: to wit a construction by 
Hilbert flowing from, or expressive of, his belief in the reality of the 
axiomatic world, where he condensed - or freeze-dried - his native intuition of
space, extracted from that intuition certain parameters, such as perhaps the
potential angularity (if not orthogonality) of its frame, and compound (complex)
factors such as vectors or tensors, which extractions (or extracts) Hilbert 
then multiplied by the potential infinity of natural cardinal numbers,
to generate an infinity of potential "Hilbert Spaces" which might serve as
store-houses (Abstellräume) of the potentially infinite volume or numbers
of treasures - or garbage - which could be generated by himself and by those
of Hilbert's colleagues who endorsed Hilbert's axiomatics, Leopold Kronecker
being the most prominent excluded competitor.  

What is to me compellingly obvious: that Hilbert was unable to communicate his
conceptions, be they axioms or intuitions, [unable to find foster home for
his legitimate or illegitimate (bastard) thought] without using the word 
"space" which is the most legitimate offspring of intuition. As opposed to
"Hilbert Space", "Hilbert XXXXX" or Hilbert ?????" would have no meaning and 
would remain forever incomprehensible. 

The same (axiomatic) considerations pertain to the word "infinity". This is a 
term which Georg Cantor misused. It also has primarily an intuitive 
meaning, a meaning grasped only by intuition, a meaning only purportedly 
axiomatized by reference to iteration. What is axiomatized, however is only 
superficial, because the postulated infinity of the number of repetitions, 
is also a concept which can be apprehended only by intuition; and never by 
axiomatic construction. 
    
Axiomatization, it seems to me, is a shell game of sorts where intuition is
concealed under the veil of purportedly non-intuitive "logical" axioms,
an enterprise with which I now (foolishly) regret not having spent, (?wasted) 
my life. "Mathematics is the queen of the sciences and number theory is the 
queen of mathematics." (Gauß)  "Die Mathematik hielt Gauss um seine eigenen 
Worte zu gebrauchen, für die Königin der Wissenschaften und die Arithmetik 
für die Königin der Mathematik." überliefert in Wolfgang Sartorius von 
Waltershausen, Gauss zum Gedächtnis.