Introduction to Hilbert Space (2022) [pdf]

&gt; The name quantum in quantum theory is related to the fact that in a separable Hilbert space, any set of mutually orthonormal vectors is countable.<p>Pretty sure the name quantum comes from the fact that some physical phenomena (eg absorption spectrum) get discrete allowed values. That in principle has nothing to do with separability (eg you can come up with non separable spaces which have operators with discrete spectrum). In fact presenting things this way is pretty confusing since separable Hilbert spaces do support operators with continuous spectrum (which is not obvious!) As far as I know separability is mostly technical, and often added to make life a bit simpler, since it&#x27;s pretty hard to come up with useful non separable Hilbert spaces.

The author&#x27;s website has a treasure trove of articles explaining different things in Physics: <a href="https:&#x2F;&#x2F;www.cphysics.org&#x2F;" rel="nofollow noreferrer">https:&#x2F;&#x2F;www.cphysics.org&#x2F;</a>

Interesting fact about Hilbert spaces: The inner product of a Hilbert space induces a norm and thus every Hilbert space is a Banach space. But what about the converse? Say we only have a normed vector space, can we decide if there is an inner-product space that actually induces this norm? The answer is yes! Simply check if the Parallellogram Law [1] holds.<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Parallelogram_law" rel="nofollow noreferrer">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Parallelogram_law</a>

Q: So what is the dimension of my Hilbert Space?<p>A: Just enough to describe all your independent physical states, which is proportional to the number of your particles.<p>Q: So if some interaction creates more particles I have to increase the dimension of my Hilbert Space?<p>A: Yes, I&#x27;m afraid so.<p>Q: But I am used to physics providing some constant background to goings-on here and now. It was absolute Newtonian space and time, then it was warped Einsteinian spacetime, now you say it depends on how many particles I create?<p>A: Yes.<p>Q: But that&#x27;s insane?<p>A: Yes.<p>A: OK, I&#x27;ll allow you to create or destroy as many particles as you like, especially if you have an really big Large Hardon Collider.<p>Q: Are you sure I can keep my Hilbert Space the same?<p>A: Yes, aha, I have thought of a special number that does not change when you add or remove finite integers...<p>Q: There is no such number ... oh wait, you mean infinite dimensional?<p>A: Yes, infinite dimensional, and complex, of course.<p>Q: Of course.<p>Q: Does not sound in the least bit plausible. How do you add particles?<p>A: It&#x27;s called the second quantization of Quantum Field Theory using Fock Spaces.<p>Q: Makes perfect Focking sense.

Hmm, he lost me on page nine where &quot;complete&quot; is explained.<p>&gt; <i>Complete means that every sequence of vectors |a1&gt;, |a2&gt;, ... satisfying lim ...</i><p>How are the elements of this sequence related? And why are we only interested in the elements where the index n&#x2F;m goes to infinity? What does that even mean if the sequence is arbitrary?<p>That&#x27;s probably why I also can&#x27;t make sense of this:<p>&gt; <i>Loosely speaking, saying that a Hilbert space is complete means that it contains all of its limits.</i>

Introduction to Hilbert Space (from a physics perspective)**<p>I prefer Halmos&#x27; book.

Professor Hilbert was unaware that other mathematicians had started calling infinite dimensional inner-product spaces, Hilbert spaces. In fact some of the foundational results were proven by Von Neumann.<p>The story goes that once Neumann was giving a lecture in Germany on such spaces with Hilbert in attendance. Hilbert had supposedly raised his hands to ask, Dr. von Neumann, I would very much like to know, what after all is a Hilbert space? Hilbert was quite underwhelmed by the definition, &quot;is that all&quot; he remarked.

I&#x27;ve made several runs now at trying to understand Hilbert space from a physics perspective, but I still do not have a good intuition for it. Something about it just breaks my brain.