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August 10

Tensor product between two functions

It looks confusing in this Q&A. So which is really the tensor product of two functions $f$ and $g$ ?

$(f\otimes g)(x,y)=f(x)g(y)$
$(f\otimes g)(x,y)=f(x)\otimes g(y)$

And if the second one is the answer, what does $f(x)\otimes g(y)$ mean? - Justin545 (talk) 06:56, 10 August 2019 (UTC)[reply]

@Justin545: They are functions taking values in H, a Hilbert space. This means that we are considering the tensor product of Hilbert spaces here, and that the result of the tensor product is a function with codomain

H\otimes H

. If H is just the reals, then this is a one-dimensional tensor product, which is just the scalar product (as the first comment assumed), which can be seen from choosing the basis

{1}

and applying the definition of the tensor product. In general, your second line is what is meant, and is the function obtained by first computing the vectors f(x) and g(y) in H, then taking the tensor product of the resulting vectors.--Jasper Deng (talk) 10:21, 10 August 2019 (UTC)[reply]

So

(f\otimes g)(x,y)=f(x)\otimes g(y)

is the general answer for any Hilbert space

H

. And if

H_{R}

is the Hilbert space consists of just reals,

(f\otimes g)(x,y)=f(x)g(y)

can be the more specific answer. Is it correct? - Justin545 (talk) 11:30, 10 August 2019 (UTC)[reply]

It's usual (in math in general) to define operations on functions pointwise, so (f+g)(x) = f(x)+g(x), (f⋅g)(x) = f(x)⋅g(x), (f⊗g)(x) = f(x)⊗⋅g(x). The example in Hilbert space#Tensor products gives another possible meaning though, and I think this is what the original answer was trying to get at in StackExchange. Specifically, L²([0, 1], C), the space of L² functions from [0, 1] to the complex numbers C, is a Hilbert space. As such you can form the tensor product L²([0, 1], C)⊗L²([0, 1], C) which can be identified with (see the example) L²([0, 1]², C) via (f⊗g)(x, y)) = f(x)g(y). But the OP (in StackExchange) was talking (presumably) about functions in L²([0, 1], H) where H is a general Hilbert space. There is no multiplication H×H→H so this wouldn't work, so the second answer had (f⊗g)(x, y)) = f(x)⊗g(y) which maps L²([0, 1], H)⊗L²([0, 1], H) to a subspace of L²([0, 1], H⊗H). I don't know if this mapping is onto, which would identify L²([0, 1], H)⊗L²([0, 1], H) with L²([0, 1], H⊗H), but it does make for an alternate interpretation of f⊗g. So I think the actual meaning of f⊗g is ambiguous and a healthy teaspoon of context is needed, though Hilbert spaces aren't my usual stomping grounds and maybe the context is clear if you're more familiar with these things. --RDBury (talk) 13:02, 10 August 2019 (UTC)[reply]

The two Hilbert spaces would have different inner products, so in that regard they wouldn't be identical. As vector spaces, they should have the same dimension, though I'm not sure of that.--Jasper Deng (talk) 18:12, 10 August 2019 (UTC)[reply]

I wanted to say the concept behind the question and the related Wikipedia articles would be too abstract to understand to me. But thanks for helps, guys. - Justin545 (talk) 22:01, 10 August 2019 (UTC)[reply]

(edit conflict) I'm late to the party and rusty with this stuff, but it might be worth pointing out that what the original poster may have wanted instead was the space of square-integrable functions with values in a C*-algebra, not in a Hilbert space. (C is special because it's both). Now answer 1 does make sense (it might even be correct, but I'm not sure). –Deacon Vorbis (carbon • videos) 22:12, 10 August 2019 (UTC)[reply]

number higher than expotentials

My girlfriend and I were playfully saying who loves who more. She sent me an emoji if a kiss. I sent back two emojis. We went back and forth and finally she sent me an emoji of a kiss "infinity." Not to be outdone, I sent her an emoji "infinity to the power of infinity." My question: is there a number system that is higher than exponents? I seem to remember someone (long ago when I was in school (not paying any attention!)) saying there was. I was thinking factorials (∞!) but when I read the article, it did not seem to be what I was thinking of. With this in mind, can she love me more than "infinity to the power of infinity?" 2001:1970:5863:5C00:F5D8:8D5D:7862:58FB (talk) 23:56, 10 August 2019 (UTC)[reply]

I'm supposed to assume good faith, but I have to admit, your story did make me throw up a little in my mouth. That being said, infinity isn't a number*. There's also really no limit to how fast you can make ever-faster increasing sequences of numbers. See Conway chained arrow notation and Knuth up-arrow notation. But again, keep in mind, writing something like

\infty \uparrow \uparrow \infty

is just as meaningless as when you wrote

\infty !

or

\infty ^{\infty }.

*The fine print: There are number systems that have infinite numbers and even ones where the numbers themselves are simply the possible sizes of infinite sets, but that's getting a bit out of scope here.

–Deacon Vorbis (carbon • videos) 00:47, 11 August 2019 (UTC)[reply]

See Tetration, but note that no matter what meaning you give your mathematical symbols, at the end of the day you can only have manipulated a finite number of symbols using a finite number of rules. This means that any mathematical system can always be re-interpreted as a finitistic discrete mathematical system, even if the intended interpretation invoked very large infinite quantities. Count Iblis (talk) 00:42, 11 August 2019 (UTC)[reply]

User:Count Iblis's claim is either trivial or false, depending on how you read it. If you take the formalist view that mathematics is not really about anything, that it's all analytic, just spinning linguistic utterances that if true are true by definition, then the claim is trivial; it boils down to saying that, because you can say only finitely many things, you can say only finitely many things.

If on the other hand it's a claim that every theory has a finite model, well, that's false. --Trovatore (talk) 17:29, 15 August 2019 (UTC)[reply]

I know how my story sounded and at my age I wanted to throw up too! But, it is what it is. My query is in good faith! I am aware that infinity is not a number; it was just part of the playfulness. Thanks. Much appreciated. — Preceding unsigned comment added by 2001:1970:5863:5C00:E082:FDC6:EEDD:9923 (talk) 17:56, 11 August 2019 (UTC)[reply]

One obvious thought is that there are different kinds of infinity. You can begin to get a feel for this by looking at our article on cardinality. In particular, you'll see that there are a whole sequence of "infinities", starting with

\aleph _{0}

(read it as "Aleph-zero"}, and followed by

\aleph _{1}

,

\aleph _{2}

,

\aleph _{3}

and so on, with

\aleph _{0}<\aleph _{1}<\aleph _{2}<\ldots .

I should say that this is just a very simple glimpse of a complicated branch of mathematics/philosophy/logic - the study of transfinite numbers. But the fact that there are all these different "infinities" may be enough to entertain you and your girlfriend. RomanSpa (talk) 01:00, 12 August 2019 (UTC)[reply]

Yes. But I don't think we can really do just to the question here.

If you want to learn about this beyond just playful curiosity, I recommend studying about Ordinal numbers and Ordinal arithmetic. It's a beautiful, elegant theory with many applications; its arithmetic is to a large part familiar; and it contains infinite numbers. Also importantly, it is a key part of many of the other things I will suggest.

There are also Cardinal numbers, but there is less you can do with those without really going deep into the theory. Also, all cardinal numbers are

\aleph _{n}

for some ordinal number n, so understanding ordinal numbers is crucial if you want to learn about cardinal numbers.

And there is also the study of finite, but absolutely ginormous, natural numbers - called Googology. It features many ways to construct large numbers - exponentiation is really not even touching the surface. You have tetration, up-arrow notation and chained arrow notation (where each method can easily express numbers which would take a whole universe to express with the previous method in the list)... And after you're done warming up, there are ways to match every (countable) infinite ordinal number with a finite number, allowing you to express numbers so stupidly large it's not even funny. Some of these methods are listed here. -- Meni Rosenfeld (talk) 15:41, 12 August 2019 (UTC)[reply]

You might also like Cantor's attic. 173.228.123.207 (talk) 11:12, 17 August 2019 (UTC)[reply]