[HN Gopher] Hilbert space: Treating functions as vectors
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Hilbert space: Treating functions as vectors
Author : signa11
Score : 134 points
Date : 2025-11-13 10:41 UTC (8 days ago)
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| petesergeant wrote:
| > But we can take it even further; what if we allow any real
| number as an index?
|
| How can an uncountably infinite set be used as an index? I was
| fine with natural numbers (countably infinite) being an index
| obv, but a real seems a stretch. I get the mathematical
| definition of a function, but again, this feels like we suddenly
| lose the plot...
| hodgehog11 wrote:
| We do it all the time. An index is just indicative that there
| is a mapping (a function), usually from the integers. However
| we don't use the subscript notation when indexing by a
| continuum due to the discomfort you describe.
|
| The point is that we need some way to deal with objects that
| are inherently infinite-dimensional.
| eucyclos wrote:
| I'm probably ignorant of how indexes work at a nuts-and-bolts
| level, but intuitively this seems like a good idea for certain
| situations. E.g if we want to keep entries in a specific order
| but don't know ahead of time how many entries will be added
| between two existing ones. House numbers in areas with a lot of
| development are an example of the kind of problem this seems
| ideal to solve, when there's a clear 'order' based on geography
| but no clear limit on the number of addresses that could be
| added 'between' existing addresses.
| ncfausti wrote:
| That's kind of how I understand it as well.
| codebje wrote:
| I think you're still describing a countably infinite set:
| there's a bijection between the natural numbers and the set
| of houses.
|
| One way to think about it is that, even though you're
| defining an index that permits infinite amounts of
| subdivision, from any given house there's always a "next
| house up" in the vector: you can move up one space.
|
| In a real-indexed vector, that notion doesn't apply. It's
| "infinity plus one" all the way down: whatever real value you
| pick to start with, x, there's no delta small enough to add
| to it such that there's no number between x and x+d.
| mb7733 wrote:
| > In a real-indexed vector, that notion doesn't apply. It's
| "infinity plus one" all the way down: whatever real value
| you pick to start with, x, there's no delta small enough to
| add to it such that there's no number between x and x+d.
|
| Just to clarify, uncountability isn't necessary for this.
| It's true for the rational numbers too, which are
| countable.
| seanhunter wrote:
| Yes. Indexes in infinite sets are counterintuitive, and
| real numbers even more so.
|
| The famous counterexample to all of this sort of thinking
| is Hilbert's hotel, which I'm sure you know but want to
| point it out for people who haven't seen it before
| because it's pretty mind-blowing when you first encounter
| it.
|
| Say you have a hotel with an infinite number of rooms
| numbered 1,2,3,... and so on and they are all occupied. A
| guest arrives- how do you accommodate them? Well you ask
| the person in room one to move to room 2, the person in
| room 2 to move to room 3, and in general the person in
| room n to move to room n+1. So every existing guest has a
| room and room 1 is now free for your new guest.
|
| Ok but what if an infinite number of prospective guests
| arrive all at once and every room in your hotel is full.
| How do you accommodate them? Still no problem. You ask
| the guest in room 1 to move to room 2, the one in room 2
| to move to room 4, and in general the guest in room n to
| move to room 2n. Now all your existing guests still have
| a room but you have freed up an infinite number of (odd-
| numbered) rooms for your infinite number of new guests to
| move into.
|
| These are all countable infinities, and Cantor showed
| that if the number of rooms in your infinitely-roomed
| hotel is _0, then the number of real numbers is 2^_0,
| which is obviously quite a lot more.
| meindnoch wrote:
| c.f.: fractional indexing
| sorokod wrote:
| The author is stretching an analogy, it's a price to pay for
| starting with R^3 as a motivational example. There is nothing
| in the general definition of a vector space that requires it's
| elements to be "indexed"
| ncfausti wrote:
| I think that's why the author put "vector" in quotes. I kind of
| imagine it as an ephemeral, infinite list where for some real,
| when we use that real value as an index into our
| "vector"/function, we get the output value as the item in this
| infinite, ephemeral list.
|
| I think the only thing that matters is that the indices have an
| ordering (which the reals obviously do) and they aren't
| irrational (i.e. they have a finite precision).
|
| Imagine you have a real number, say, e.g. 2.4. What stops us
| from using that as an index into an infinite, infinitely
| resizable list? 2.4^2 = 5.76. Depending on how fine-grained
| your application requires you could say 2.41 (=5.8081) is the
| next index OR 2.5 (=6.25) is the next index we look at or care
| about.
|
| I could be misunderstanding it, though.
| a57721 wrote:
| A vector is always a vector -- an element of something that
| satisfies the axioms of a vector space. The author starts
| with the example of R^n, which is a very particular vector
| space that is finite-dimensional and comes with a "canonical"
| basis (0,...,1,...,0). In general, a basis will always exist
| for any vector space (using the axiom of choice), but there
| is no need to fix it, unless you do some calculations. The
| analogy with R^n is the only reason the "indices" are
| mentioned, and I think this only creates more confusion.
|
| > and they aren't irrational (i.e. they have a finite
| precision)
|
| No, if you want only rational "indices", then your vector
| space has a countable basis. Interesting vector spaces in
| analysis are uncountably infinite dimensional. (And for this
| reason the usual notion of a basis is not very useful in this
| context.)
| Natsu wrote:
| > and they aren't irrational (i.e. they have a finite
| precision).
|
| I'm not sure if I'm misunderstanding what you mean by 'finite
| precision' but the ordinary meaning of those words would seem
| to limit it to rational numbers?
| zozbot234 wrote:
| In practice you're always computing with finite precision.
| (Even computing with symbolic expressions is just a
| preliminary step to what's ultimately a numerical result
| with finite precision.) The whole point of real numbers is
| to abstract away from detailed considerations of precision,
| and figure out what happens if you only ever care about
| putting satisfactory bounds on the output and are willing
| to bound your input to the extent required.
| codebje wrote:
| The only difference of note, I think, is that you can't
| enumerate the elements. Instead of being able to say "for each
| element, ..." you'd have to say "for all elements, ...", like
| the example of vector length defined as an integral over the
| full number range.
| Jaxan wrote:
| To a mathematician "each" and "all" are synonyms.
| drdeca wrote:
| What do you understand "index" to mean here? To me, a family
| indexed by some set is mostly just a different notation for,
| and attitude towards, a function with domain the indexing set.
| majikaja wrote:
| Consider a function on R as an |R|-dimensional vector...
| ttoinou wrote:
| Which defeats the purpose of thinking about functions as a
| vector space. It's all smokes and mirrors
| jhanschoo wrote:
| You can look at use cases for an index, and see how well they
| hold up.
|
| Asking where the smallest greater number (next number) is no
| longer makes sense.
|
| Taking two numbers and asking whether one is greater than the
| other still makes sense. (and hence also whether they are
| equal)
|
| Taking two numbers and asking how far separated from each other
| still makes sense.
|
| You may already observe some uses for indexes in programming
| that don't use all of these properties of an index. For
| example, the index of a hash set "only cares about equality",
| and "the next index" may be an unfilled address in a hash set.
| IronyMan100 wrote:
| This Has it's use. The continouus Fourier Transform is is based
| on that. You are asking what frequencies is this continouus
| signal made of. Time is normally defined as a real number in
| that context, but If you have a continouus time you need
| continouus frequencies to map time space to frequency space.
| You can think about an Index as a lego Block, that you need to
| construct Something.
| hyghjiyhu wrote:
| I think getting hung up on words (in this case index) in
| mathematics is a trap. They are often stretched to their
| breaking point and you just kind of go with the flow.
|
| > When I use a word,' Humpty Dumpty said in rather a scornful
| tone, 'it means just what I choose it to mean -- neither more
| nor less.'
|
| > 'The question is,' said Alice, 'whether you can make words
| mean so many different things.'
|
| > 'The question is,' said Humpty Dumpty, 'which is to be master
| -- that's all.
|
| In mathematics it is the author's privilege.
| shiandow wrote:
| Well there's no law against it.
|
| Okay I suppose the axiom of choice is somewhat necessary to
| make it make sense. But only because otherwise such an indexed
| object may fail to exist.
|
| Anyway arbitrary indexes are useful, you often end up doing
| stuff like covering a space by finding a covering set for each
| individual point. And then using compactness to show you only
| need finitely many to cover the whole space. It is doable
| without uncountable indices, but it makes it very difficult to
| write down.
| super_mario wrote:
| In ZFC set theory, indexed family over a set (possibly
| uncountable or even bigger), is just syntactic sugar for a
| function.
|
| So let's say you have a set U (possibly uncountable). To say
| let {u_i}, i in I (another set, possibly uncountable) is
| equivalent to asserting existence of function f:I -> U, such
| that f(i) = u_i. Note that this does not even require axiom of
| choice, since you are allowed to postulate that a function
| exists.
|
| Of course if I is uncountable you can't list the elements of I,
| but that is not important in this case.
| constantcrying wrote:
| As evidenced by the confusion of at least one commenter, I do not
| think it is a good didactic way to introduce vectors by how they
| can be written in a particular basis.
|
| It is just unhelpful in many ways. It fixates on one particular
| basis and it results in a vector space with few applications and
| it can not explain many of the most important function vector
| spaces, which are of course the L^p spaces.
|
| In most function vector spaces you encounter in mathematics, you
| can not say what the value of a function at a point is. They are
| not defined that way.
|
| The right didactic way, in my experience, is introducing vector
| spaces first. Vectors are elements of vector spaces, not because
| they can be written in any particular basis, but because they
| fulfill the formal definition. And because they fullfil the
| formal definition they can be written in a basis.
| lanza wrote:
| > It fixates on one particular basis and it results in a vector
| space with few applications and it can not explain many of the
| most important function vector spaces, which are of course the
| L^p spaces.
|
| Except just about all relevant applications that exist in
| computer science and physics where fixating on a representation
| is the standard.
| constantcrying wrote:
| Most relevant applications use L^2 spaces which can not be
| defined point wise.
|
| If you want to talk about applications, then this
| representation is especially bad. Since the intuition it
| gives is just straight up false.
| defmacr0 wrote:
| Fwiw, my favourite textbook in communication theory
| (Lapidoth, A Foundation in Digital Communication)
| explicitly calls out this issue of working with equivalence
| classes of signals and chooses to derive most theorems
| using the tools available when working in _2 (square-
| integrable functions) and _1 space
| rnhmjoj wrote:
| In physics it is common to work explicitily with the
| components in a base (see tensors in relativity or
| representation theory), but it's also very important to
| understand how your quantities transform between different
| basis. It's a trade-off.
| arjie wrote:
| Haha, this works if you already know what a vector space is.
| But I think pedagogy needs to provide motivating examples. I'll
| quote one section of a text by Poincare (translated by an LLM
| since most here do not speak French).
|
| > _We are in a geometry class. The teacher dictates: "A circle
| is the locus of points in the plane that are at the same
| distance from an interior point called the center." The good
| student writes this sentence in his notebook; the bad student
| draws little stick figures in it; but neither one has
| understood. So the teacher takes the chalk and draws a circle
| on the board. "Ah!" think the students, "why didn't he say
| right away: a circle is a round shape -- we would have
| understood."
|
| > No doubt, it is the teacher who is right. The students'
| definition would have been worthless, since it could not have
| served for any demonstration, and above all because it would
| not have given them the salutary habit of analyzing their
| conceptions. But they should be shown that they do not
| understand what they think they understand, and led to
| recognize the crudeness of their primitive notion, to desire on
| their own that it be refined and improved._
|
| The learning comes from making the mistake and being corrected,
| not from being taught the definition, I think.
|
| Anyway, it's from Science and Method, Book 2
| https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre...
|
| There's more to the section that talks about the subject. I
| just find this particular paragraph amusingly germane.
| constantcrying wrote:
| I have nothing against starting out with motivating examples,
| obviously they are needed for understanding. But they should
| motivate the definition of a vector space. Not the definition
| of vectors as mappings of indices.
|
| Functions are actually a great motivating example for the
| definition of a vector space, precisely because they are
| first look nothing like what student think of as a vector.
| arjie wrote:
| Thinking about this specific case, I think you are right.
| The manner of describing actually confuses the concept more
| than if it never tried to introduce the index-mapping.
| omnicognate wrote:
| It's trivial to provide motivating examples for vector
| spaces, and there's no reason you can't do so while
| explaining what they actually are, which is also very simple
| for anyone who understands the basic concepts of set,
| function, associativity and commutativity. The notion of a
| basis falls out very quickly and allows you to talk about
| lists of numbers as much as you like without ever implying
| any particular basis is special.
|
| I hesitate to call anything pedagogically "wrong" as people
| think and learn in different ways, but I think the coyness
| some teachers display about the vector space concept hampers
| and delays a lot of students' understanding.
|
| Edit: Actually, I think the "start with 'concrete' lists of
| numbers and move to 'abstract' vector spaces" approach is
| misguided as it is based on the idea that the vector space is
| an abstraction of the lists of numbers, which I think is
| wrong.
|
| The vector space and the lists of numbers are two equivalent,
| related abstractions of some underlying thing, eg. movements
| in Euclidean space, investment portfolios, pixel colours,
| etc. The difference is that one of the abstractions is more
| useful for performing numerical calculations and one better
| expresses the mathematical structure and properties of the
| entities under consideration. They're not different levels of
| abstraction but different abstractions with different uses.
|
| I'd be inclined to introduce the one best suited to
| understanding first, or at least alongside the one used for
| computations. Otherwise students are just memorising
| algorithms without understanding, which isn't what maths
| education should be about, IMO. (The properties of those
| algorithms can of course be proved without the vector space
| concept, but such proofs are opaque and magical, often using
| determinants which are introduced with no better
| justification than that they allow these things to be
| proved.)
| zapw wrote:
| For many students, it is not so simple to grasp the concept
| of an abstract vector space. They could be taking linear
| algebra as college freshmen, without having seen any formal
| algebraic structures before. Many are unfamiliar with the
| formal notion of a set (and certainly have not seen the
| actual axioms of a set before). Most linear algebra
| students are not actually math majors; they are typically
| studying engineering, computer science, or some other
| physical science. Examples of abstract vector spaces are
| most often function spaces of some form (for example,
| polynomials of at most a given degree). These examples are
| not so motivating for non-math students.
|
| The main reason why people care about linear algebra is
| that it lets you solve linear systems of equations (and
| perform related operations, such as projections). A linear
| system of equations has an immediate correspondence with a
| matrix of coefficients, a right-hand side vector, and a
| solution vector. For this reason, it is very natural to
| first talk about matrices and vectors (they can be used to
| represent concretely a linear system of equations), and
| then introduce the concept of vector space in cases where
| the abstract view can be clarifying or help with
| understanding.
|
| From my perspective, the "right" way to teach linear
| algebra depends on the mathematical maturity of the
| students. If they are honors math majors, they can easily
| handle the definition of an abstract vector space right
| away. If they have less mathematical maturity, the abstract
| viewpoint isn't helpful for them (at least not without
| first familiarizing themselves with the more concrete
| concepts). Think about it this way: we don't teach school
| children about natural numbers and arithmetic by first
| listing the Peano axioms.
| omnicognate wrote:
| I think at least in the UK the lack of "mathematical
| maturity" among early undergraduates is partly the result
| of this very coyness about mathematical concepts.
| Enormous time at A-Level is spent rote learning
| algorithms, and very little on grasping the basic
| concepts of mathematics, so it's hardly surprising
| students turn up unprepared for such simple notions as
| "vector space".
|
| I don't have first hand experience of the French system,
| but from what I understand the approach there is more
| along the lines I'm thinking of, and the relative over-
| representation of French graduates among my more
| mathematical colleagues suggests it may be rather
| effective in practice.
| vpribish wrote:
| That's awful - just an awful way to teach. It's from more
| than a century ago when the point was to tame the children
| and turn them into good Prussian soldiers.
|
| You don't have to start with anxiety, shame, and dominance -
| you can start with curiosity, a base of common understanding,
| and then experiment and problem solving.
| constantcrying wrote:
| >It's from more than a century ago when the point was to
| tame the children and turn them into good Prussian
| soldiers.
|
| If you judge by the outcome, that is probably the greatest
| education system of all time.
|
| >You don't have to start with anxiety, shame, and dominance
| - you can start with curiosity, a base of common
| understanding, and then experiment and problem solving.
|
| You can. The kids will learn nothing though.
|
| School nowadays is a joke. An absolute waste of time. In a
| single semester of rigorous mathematics I learned more than
| in years in school. It is cruel to waste childrens time
| like that.
|
| School needs to be authoritarian, rigorous and selective.
| lvncelot wrote:
| Completely agree. In uni, I (re)-learned about vectors in
| linear algebra, and for a good chunk of the course, we didn't
| write anything in "standard vector notation". We learned about
| vector axioms first, and then vectors were treated as "anything
| that satisfies the vector axioms". (When doing more practical
| examples, we just used the reals instead of something like R^3,
| but the entire time it was clear that for any proof, anything
| that can be added and multiplied in the way that the vector
| axioms describe would fit.) I think adopting this structuralist
| view really helps with a lot of mathematical studies.
| phkx wrote:
| Now I'm thinking that I have missed the point of the article. I
| didn't read it as an introduction to vector spaces, but rather
| that the introduction served as to give an intuition how
| functions may be viewed as vectors (going back to the article,
| it's even in the section heading). I found the next parts well
| written and to the point, leading along the steps to show that
| indeed the requirements for a Hilbert space are met by L^2
| (even though those requirements are only spelled out in the
| end). I'm not actively working with mathematics any more, but I
| didn't notice any major corner cutting. It's not text book
| rigorous but lays out the idea in an easy to follow way. I took
| something away from it - or not, depending on whether I missed
| some inconsistency.
| griffzhowl wrote:
| > In most function vector spaces you encounter in mathematics,
| you can not say what the value of a function at a point is.
|
| Could you spell out what you mean by that? Functions are all
| defined on their domains (by definition)
|
| Are you referring to the L^p spaces being really equivalence
| classes of functions agreeing almost everywhere?
| rnhmjoj wrote:
| Yes, the L^p spaces are not vector spaces of functions, but
| essentially equivalent classes of functions that give the
| same result in an Lebesgue integral. For these reason, common
| operations on functions, like evaluating at a point or taking
| a derivative are undefined.
|
| If you care about these you need something more restrictive,
| for example to study differential equations you can work in
| Sobolev spaces, where the continuity requirement allows you
| to identify an equivalent class with a well-defined function.
| griffzhowl wrote:
| Thanks for the clarification
| anuramat wrote:
| reminded me of "tensor is a bunch of numbers that transform in
| a certain way"; this should be illegal to teach, especially in
| physics
| dellasera wrote:
| Hey thanks for writing this article. I found it to be really
| great.
| vineethy wrote:
| Wish I read this years ago
| perihelions wrote:
| Tangential question that's been nagging me; in spaces of
| _functions defined pointwise_ , is there a natural way to lift
| algebraic structures (group/ring/whatever) from the base set up
| to the function space? It's generally true across many separate
| objects in math, but I don't know a clear explanation of why.
| (Category theory, I assume).
|
| Many of the propositions in the author's Appendix A are of this
| form.
|
| I.e., if you look at how addition on function spaces is defined
| pointwise, (f+g)(x) = f(x)+g(x) -- that's different meanings of
| (+) on either side -- that looks exactly like the defining
| relation of a group homomorphism, except that the symbols are
| backwards.
| dboreham wrote:
| ianam but I'm not sure there needs to be a "why" -- these
| things are defined to have certain properties, and so they have
| those properties. That's the purpose for which they were
| defined.
| samasblack wrote:
| You are observing the evaluation map ev_p: C(M, R) -> R,
| ev_p(f) = f(p), is a ring homomorphism. In that spirit, you
| might find this Terry Tao article on the Yondea lemma
| interesting: https://terrytao.wordpress.com/2023/08/25/yonedas-
| lemma-as-a...
| bmacho wrote:
| Let T^S = {f| f:S->T} be all functions from a source set S to a
| target structure (set + operations + relations + axioms) T.
|
| You can lift all the operations and relations from T to T^S,
| and you'll get a structure with the same type signature.
|
| Universal equations involving operations remain true when
| lifted. Therefore if T is a variety[0], T^S is a variety of the
| same type.
|
| So for example if S a set with 2 elements, then T^S is TxT +
| lifted properties. If T is an Abelian group then TxT is also an
| Abelian group. If T is a ring, TxT is also a ring. If T is a
| field, TxT is not a field since (0,1) has no inverse.
|
| What about the relations, what types of identities remain true
| when lifted form T to T^S?
|
| [0] : https://en.wikipedia.org/wiki/Variety_(universal_algebra)
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