[HN Gopher] Introduction to Hilbert Space (2022) [pdf]
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Introduction to Hilbert Space (2022) [pdf]
Author : azeemba
Score : 86 points
Date : 2023-09-15 15:04 UTC (7 hours ago)
(HTM) web link (www.cphysics.org)
(TXT) w3m dump (www.cphysics.org)
| queuebert wrote:
| I'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.
| antoine-levitt wrote:
| > 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.
|
| 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's pretty hard to come up with
| useful non separable Hilbert spaces.
| aquafox wrote:
| 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.
|
| [1] https://en.wikipedia.org/wiki/Parallelogram_law
| 3abiton wrote:
| This makes me want to lrarn more about hilbert space.
| sgdpk wrote:
| Yet another fact discovered by von Neumann.
| queuebert wrote:
| https://imgflip.com/i/7zaxoy
| ezion wrote:
| Introduction to Hilbert Space (from a physics perspective)**
|
| I prefer Halmos' book.
| Paul-Craft wrote:
| Do you mean _Introduction to Hilbert space and the theory of
| spectral multiplicity?_
|
| Here's a link:
| https://archive.org/details/introductiontohi0000halm
| azeemba wrote:
| The author's website has a treasure trove of articles explaining
| different things in Physics: https://www.cphysics.org/
| anta40 wrote:
| Nicely formatted PDFs there. Bookmarked :D
| rhymer wrote:
| Agree, this is great! I wonder if there're some ML equivalent
| sites that present topics in a modular way?
| esafak wrote:
| https://course.fast.ai/
|
| https://www.deeplearningbook.org/
| _a_a_a_ wrote:
| Does https://mlbook.explained.ai/ help?
|
| Been on HN before, got very positive comments. From the
| author of ANTLR no less.
| smokel wrote:
| Perhaps not exactly what you are looking for, but MLU-Explain
| is nice: https://mlu-explain.github.io/
| mh-cx wrote:
| Hmm, he lost me on page nine where "complete" is explained.
|
| > _Complete means that every sequence of vectors |a1 >, |a2>, ...
| satisfying lim ..._
|
| How are the elements of this sequence related? And why are we
| only interested in the elements where the index n/m goes to
| infinity? What does that even mean if the sequence is arbitrary?
|
| That's probably why I also can't make sense of this:
|
| > _Loosely speaking, saying that a Hilbert space is complete
| means that it contains all of its limits._
| btilly wrote:
| Putting that into plain English, "If it looks like a sequence
| is converging, it really is converging to something."
|
| What does it mean to say that the sequence looks like it is
| converging?
|
| Naively, it means that any two elements far in the sequence are
| always very close together. We make that intuition more precise
| by turning it into a challenge-response, "You tell me how close
| you want the elements to be, I'll tell you how far out to pick
| your elements." And then we write that mathematically as
| [?] [?] > 0 # You tell me how close
| [?] N # I'll tell you how far
| such that if N < n, m # so that any 2 elements
| then || a_n - a_m || < [?] # will always be that close
|
| Others have given examples about things like the rational
| numbers. The canonical example of why it matters comes from
| Fourier series. Joseph Fourier discovered this one. Consider a
| bunch of functions f(x) over the interval from 0 to 2p. We can
| create a dot product with f*g equaling the integral from 0 to
| 2p of f(x)g(x). And the length of a function f is sqrt(f*f)
|
| Joseph Fourier discovered that the following functions are
| orthonormal (each has length 1, their dot product with each
| other is 0). 1/sqrt(2p),
| sin(x)/sqrt(p), sin(2x)/sqrt(p), sin(3x)/sqrt(p), ...
| cos(x)/sqrt(p), cos(2x)/sqrt(p), cos(3x)/sqrt(p), ...
|
| (I hope I have the constants correct...)
|
| Given any function, he could compute a series that we now call
| the Fourier series that looks like it added up to that
| function. But there were complications. It added up perfectly
| for smooth functions with the same value at the edges like
| x(2p-x). But it also added up except at a few points for things
| like square waves. This caused a crisis in mathematics because
| at the time square waves were not considered functions, and
| nobody had ever realized that adding up infinite series of
| smooth functions could do such weird things. Their idea of
| functions was essentially what we would call analytic functions
| today. Basically things that look like power series. And Mr.
| Fourier had just shown that the analytic functions are not
| complete.
|
| In addition to revealing problems in how we understood math,
| his technique was very, very useful. Because now you just had
| to figure out the physics of how, say, heat spreads out or
| vibrating strings vibrate just for those those sin and cos
| terms, and then you could figure out heat and vibration for ANY
| function.
|
| Resolving the math problems started many decades of research.
| The results of which included better definitions of the real
| numbers, the [?]-d definition of a limit (or [?]-N for a
| series), new theories of integration, and the idea of Hilbert
| spaces.
| azeemba wrote:
| It helped me to look up an example of a space that is not
| "complete".
|
| Turns out, rational numbers are the classic example of a space
| that is not complete. A sequence of rational numbers can
| approximate pi but pi itself doesn't exist in the space (since
| its irrational). So the rational numbers that get closer and
| closer to pi form a limit to a value that's not in the space.
| Paul-Craft wrote:
| That's a great example. To make it concrete, you can take the
| sequence such that $a_n$ is the $n$-th partial sum of any of
| the series here that involve only rational numbers: https://e
| n.wikipedia.org/wiki/List_of_formulae_involving_%CF...
| multiplied by an appropriate constant.
| stracer wrote:
| This is standard mathematical analysis. Infinite sequence of
| elements may look like it converges to some target element,
| judging by mutual distances converging to zero. Such sequence
| is a Cauchy sequence. When the target element actually exists,
| then the sequence is also convergent. A space where every
| Cauchy sequence is convergent, is called complete.
|
| Example: if the space is all real numbers except 0, then any
| sequence of real numbers accumulating around 0 (for howsoever
| small a distance, there is always infinite number of points
| closer to 0), the sequence is a Cauchy sequence, but not
| convergent (because 0 is not present). So that space is not
| complete (has a hole).
|
| If the space is all real numbers, then the same sequence is
| also convergent, and the space is complete (no holes).
| l33t7332273 wrote:
| > How are the elements of this sequence related
|
| They are not necessarily related
|
| > And why are we only interested in the elements where the
| index n/m goes to infinity?
|
| If the sequence is finite, then we don't really care to discuss
| the "limit" of the sequence.
|
| > Loosely speaking, saying that a Hilbert space is complete
| means that it contains all of its limits.
|
| For a set S to not contain all of its limits means you can have
| an infinite sequence of points (a_n) where each a_n is in S and
| there is no point a in S so that the sequence is eventually as
| close to a as you'd like.
|
| More formally, there does not exist a in S so that for any e >
| 0 we can pick an M so that m > M implies | a_m - a | < e.
|
| You can see how "m > M" gives a formal meaning to "eventually,"
| and "for any e > 0 ... | a_m - a | < e" gives a formal meaning
| to "as close to a as you'd like."
| steppi wrote:
| Agreed that this is pretty terse. The sequences they're talking
| about are called Cauchy sequences [0]. A sequence a_i is Cauchy
| if for any epsilon, there exists an N such that if m and n are
| both greater than N, then |a_m - a_n| < epsilon. A classic
| example, suppose your space is the set of rational numbers, and
| consider the sequence a0 = 1, a_1 = 1.4, a_2 = 1.41, ... a_n =
| sqrt(2) up to n digits after the decimal place. You can verify
| that this is a Cauchy sequence, successive points get
| arbitrarily close to each other. This means the rational
| numbers are incomplete, because this Cauchy sequence of
| rationals doesn't converge to a rational number. It's the real
| numbers that forms a complete space.
|
| Completeness is required for nice results like the spectral
| theorem for self-adjoint operators [1] to hold, which is pretty
| essential for Quantum Mechanics.
|
| [0] https://en.wikipedia.org/wiki/Cauchy_sequence.
|
| [1] https://en.wikipedia.org/wiki/Spectral_theorem
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