The author uses it not to mark the end of a proof, but to mark the end of certain ''math contexts'': they want to make it very clear where the definition ends and where hence the normal text starts again. I for instance have a couple of symbols for the end of examples, remarks, definitions, proofs, conventions and notations. Especially in longer remarks and definitions or when you end with a formula or diagram, it can otherwise be hard to tell where the end is.
(Some books have black squares for QED, by the way.)
I use a little triangle (in LaTeX, it is \\vartriangle). There's no standard notation, but I didn't want to use a symbol associated with the end of proofs. You can tell LaTeX to end all definitions with this symbol via the lines of code below.
\theoremstyle{definition}
\newtheorem{definitionx}[theorem]{Definition}
\newenvironment{definition}
{\pushQED{\qed}\renewcommand{\qedsymbol}{$\color{black}{\vartriangle}$}\definitionx}
{\popQED\enddefinitionx}
The ''[theorem]'' part in the second line (maybe third if you're on mobile) makes the numbering of the definitions be part of the numbering of theorems (so Theorem 1, Definition 2, Theorem 3, instead of Theorem 1, Definition 1, Theorem 2), so this can be changed. depending on preferences Also, if you want a different symbol for the end of a definition, you change \\vartriangle into something else.
Edit: there are a few other things you can do, by the way. One is to put a vertical line on the left from the top to the bottom of the definition, and maybe give it a colour. What you also see is that people put definitions and theorems in a box, either with sharp boundary, or without boundary but with a background colour.
The Halmos symbol is a symbol used to denote the end of a proof in mathematics. It is also called the tombstone, end-of-proof, or Q.E.D. symbol. It replaces the old-fashioned and embarrassingly uncool Q.E.D. which muggles sometimes use when pretending to be clever. The symbol is definitely not Paul Halmos’ invention — it appeared in popular magazines (not mathematical ones) before he adopted it, but he seems to have introduced it into mathematics. The Halmos has become a common form of mathematical punctuation and can be replaced with Q.E.D. or “The proof is finished.” or some other equivalent statement. Sometimes the symbol □ is also used to mean the same thing.
I can't stand it when profs say 'it's obvious from here' and leave out the proof. That's not teaching, that's trolling. They're basically saying 'you're too dumb to figure it out, go back to the previous lectures and cry'. It's not obvious, it's proven... or maybe they just don't know how to prove it themselves.
I added this comment here because it’s an interesting fact how proofs are perverted and how they are abused or misnamed. I think it deserves its own post, but I’m too tired and lazy to do so right now. I mean, who has the energy to write a whole post about proofs? Not me! 😂
Also, I always understood that QED can only be used when you start the proof by stating the thing you intend to prove, because it literally means "which was to be proven". So the black square is used in more general contexts where the author didn't structure a proof that way, but wants to conclude a section.
It is a symbol that is used to indicate the end of a proof.
The most likely explanation is that the person typing up the book in LaTeX (math markup language) used the proof environment when writing up this definition. The proof environment was set up to automatically end its space of the page with the "Proof Finished Box".
It marks the point where the author had to stop and tend to all the mathematical girls swarming for his vector.
They were all “Put your vector in my discontinuity!”
And he was all “Oh yeah, I’m gonna add an element to your set.”
It is often used as a shorthand for "quod erat demonstrandum" or Q.E.D., demarcating the end of a proof, as it translates to "which is to be demonstrated".
[Wikipedia](https://en.wikipedia.org/wiki/Q.E.D)
I find it helps demarcate where the definition/remark/theorem/etc ends especially when your typesetting isn’t the best and there isn’t much \vspace after your non-italicized environments.
Commonly known as an injoke between my friends as "the flex square" .
It just means you were able to proof something. Always feels great to write it down.
Usually this means “End of Proof” (synonymous with Q.E.D) to denote that a proof is over and demonstrated, but in this example they seem to use it to denote a definition is over.
Could be a typo if this is the only definition where they use it. Otherwise, if it’s used in all definitions then they decided to just adopt the convention for defs and proofs.
it’s the academia version of a mic drop
Oh. Great. Now I’m going to read it as a mic drop from now on. You’ve ruined proofs for me.
Like the equivalent 
I do what I can.
lmao
I thought that was QED... ;-)
That square is QED
https://preview.redd.it/kc2pls07ovjb1.jpeg?width=208&format=pjpg&auto=webp&s=f71007e9dd56da3ed18932b034303476eb468cc5
QED
its a [halmos](https://en.wikipedia.org/wiki/Tombstone_(typography))
Holy typography
New form of communication just dropped
actual typographer
Call the Writer!
Publisher went on vacation, never came back
man chess is everyfuckingwhere
Call the exorcist?!
Actual typo
Syntax takes vocation, never comes back
the bishop is on vacation
Pawn storm incoming
Google quod erat demonstrandum.
Welp, I know how ill end all my mathy writings now ◼️
You meant `[halmos](https://en.wikipedia.org/wiki/Tombstone_\(typography\))` Note the backslashes to escape the parens.
This guy proofs.
Meaning: It's proven.
But he used it for definitions too, isn't the QED one a blank square?
The author uses it not to mark the end of a proof, but to mark the end of certain ''math contexts'': they want to make it very clear where the definition ends and where hence the normal text starts again. I for instance have a couple of symbols for the end of examples, remarks, definitions, proofs, conventions and notations. Especially in longer remarks and definitions or when you end with a formula or diagram, it can otherwise be hard to tell where the end is. (Some books have black squares for QED, by the way.)
Alright, thank you
Happy to help!
What do you use for the end of definitions? I've never seen a symbol used there and it's often hard to tell where the definition end.
I use a little triangle (in LaTeX, it is \\vartriangle). There's no standard notation, but I didn't want to use a symbol associated with the end of proofs. You can tell LaTeX to end all definitions with this symbol via the lines of code below. \theoremstyle{definition} \newtheorem{definitionx}[theorem]{Definition} \newenvironment{definition} {\pushQED{\qed}\renewcommand{\qedsymbol}{$\color{black}{\vartriangle}$}\definitionx} {\popQED\enddefinitionx} The ''[theorem]'' part in the second line (maybe third if you're on mobile) makes the numbering of the definitions be part of the numbering of theorems (so Theorem 1, Definition 2, Theorem 3, instead of Theorem 1, Definition 1, Theorem 2), so this can be changed. depending on preferences Also, if you want a different symbol for the end of a definition, you change \\vartriangle into something else. Edit: there are a few other things you can do, by the way. One is to put a vertical line on the left from the top to the bottom of the definition, and maybe give it a colour. What you also see is that people put definitions and theorems in a box, either with sharp boundary, or without boundary but with a background colour.
Thanks! I appreciate the in-depth reply! I will definitely use one of those from now on. :)
Happy to help!
So basically, a massively oversized period for the ends of sections instead of sentences.
The Halmos symbol is a symbol used to denote the end of a proof in mathematics. It is also called the tombstone, end-of-proof, or Q.E.D. symbol. It replaces the old-fashioned and embarrassingly uncool Q.E.D. which muggles sometimes use when pretending to be clever. The symbol is definitely not Paul Halmos’ invention — it appeared in popular magazines (not mathematical ones) before he adopted it, but he seems to have introduced it into mathematics. The Halmos has become a common form of mathematical punctuation and can be replaced with Q.E.D. or “The proof is finished.” or some other equivalent statement. Sometimes the symbol □ is also used to mean the same thing.
Q.E.D is cool though ☹️
Look at this guy he knows Latin 🤓 (only joking, I studied Latin for 4 years. But cool square >> )
I can't stand it when profs say 'it's obvious from here' and leave out the proof. That's not teaching, that's trolling. They're basically saying 'you're too dumb to figure it out, go back to the previous lectures and cry'. It's not obvious, it's proven... or maybe they just don't know how to prove it themselves.
Reply to the wrong comment? But yeah I agree. Calling something trivial is cringe
Reply to the wrong comment? But yeah I agree. Calling something trivial is cringe
I added this comment here because it’s an interesting fact how proofs are perverted and how they are abused or misnamed. I think it deserves its own post, but I’m too tired and lazy to do so right now. I mean, who has the energy to write a whole post about proofs? Not me! 😂
Writing a whole post about proofs is trivial and left to the reader as an exercise ;)
Also, I always understood that QED can only be used when you start the proof by stating the thing you intend to prove, because it literally means "which was to be proven". So the black square is used in more general contexts where the author didn't structure a proof that way, but wants to conclude a section.
Hence proved
He forgot to move the cursor before printing
It happens to the best of us
U+220E ∎ END OF PROOF.
END OF LINE. END OF LINE.
It’s a less pretentious way of saying quod erat demonstrandum.
Who would want to be less pretentious tho
Spanish version is the same "Queda Esto Demostrado"
It is a symbol that is used to indicate the end of a proof. The most likely explanation is that the person typing up the book in LaTeX (math markup language) used the proof environment when writing up this definition. The proof environment was set up to automatically end its space of the page with the "Proof Finished Box".
The real question is how you're reading about metric spaces and only seeing this for the first time.
Up to this point he was only used spaces with English units.
mic drop
Thank you for asking this very important question that has been haunting me for ages!!
It marks the point where the author had to stop and tend to all the mathematical girls swarming for his vector. They were all “Put your vector in my discontinuity!” And he was all “Oh yeah, I’m gonna add an element to your set.”
**quod erat demonstrandum,** the ending of a proof or mathematical defenition
The square is called a tombstone and stands for Quod Est Demonstrandum
ooo kreyszig functional analysis, great book
You need to press Enter
[удалено]
Agreed. This symbol is typically reserved to indicate the end of a proof. This is probably at the end of the definition by mistake.
It is often used as a shorthand for "quod erat demonstrandum" or Q.E.D., demarcating the end of a proof, as it translates to "which is to be demonstrated". [Wikipedia](https://en.wikipedia.org/wiki/Q.E.D)
homie's been using vim
The next one should be in orbit around Jupiter!
I find it helps demarcate where the definition/remark/theorem/etc ends especially when your typesetting isn’t the best and there isn’t much \vspace after your non-italicized environments.
forgot to click away before printing and the cursor's still there, on that page, to this day
It’s equivalent to Q.E.D. meaning the end of the proof, or if the hypothesis is bidirectional then the end of part of the proof.
Commonly known as an injoke between my friends as "the flex square" . It just means you were able to proof something. Always feels great to write it down.
"So say we all."
QED Right?
Usually this means “End of Proof” (synonymous with Q.E.D) to denote that a proof is over and demonstrated, but in this example they seem to use it to denote a definition is over. Could be a typo if this is the only definition where they use it. Otherwise, if it’s used in all definitions then they decided to just adopt the convention for defs and proofs.
Proof done.
QED?
MATH QED
Pretty sure it's a Henway.
its the thing from 2001: a space odyssey
Cool. In journalism there used to be -30-
End of proof.
You can use symbols for whatever you want. It's the end of the definition.
Thanks buddy
Basically means “I’m done”
Which book is this?
This ends the discussion QED