Dual Spaces: The 'Underworlds' of Mathematics

Dual spaces are defined for general vector spaces, but since vector spaces themselves are an abstract concept, I have taken an approach that focuses on a simple example: vectors of real numbers. If you are interested in learning more about dual spaces, please explore vector spaces and their dual spaces.

[BGM]

[Materials]
VOICEVOX:ずんだもん (Zundamon, illustrated by sakamoto_AHR)
VOICEVOX:四国めたん (Shikoku Metan, illustrated by sakamoto_AHR)
(+ other tools)
pixabay

This video uses synthesized voices.

#math

[muhammederenduran]

I'm studying mechanical engineering and your videos explains some subjects better than most content creators dedicated to the field. Keep up the good work!

[Igorious]

«I don't understand this at all anymore!» — is my regular mood on higher math lectures 😅

[UnemptyVoid]

Im happy that at least this video's comment section refers to zudamon as vocaloid and not an anime

[신우석-nz]

韓国から来ました。私は高校生です。
この数字教室は本当に面白かった。
ありがとう Zundamon！

[AmitPrakashJena]

imagine being an undergrad student in the freshman year and getting the "dweller of the underworld" achievement.

[emiliavalentino]

Learning about a vocaloid from math is a new one. And soon, learning about math from a vocaloid will be too!

[nanigov]

Duality is one of those mathematical things that manages to rear its head into most fields in some shape or form, and it's always so cool when it shows up! I still remember being enamored by dual spaces and the canonical isomorphism to the bi-dual when I first saw them

[robharwood]

I like the presentation of the topic, using this dialogue between two cute but intelligent characters. Like a Socratic Dialogue, but in anime style!

I do have a critique/suggestion, though. I think this kind of introduction to Dual Spaces could have benefited a lot from having a much more in-depth exploration of finite-dimensional dual spaces, such as by using a thorough example.

The start of the video says, "If you want to return to the Upperworld, you have no choice but to solve this problem." And it has a nice diagram. However, that specific, concrete diagram/problem is never actually 'solved' by the end of the video. We're not even really shown any of the details of this illustrative problem. Nor given any motivation for why such a problem might be worthwhile to solve.

There are many areas where concrete dual spaces can be used for solving concrete problems, but this video kept everything quite abstract. Perhaps there could have been a simple example of Linear Programming. Or some kind of geometry example. Or something else. (Actually, I came here wanting to brush up on my understanding of Dual Spaces, so I actually don't have a lot of examples of how they can be used in math; was hoping this video would present one...)

Anyway, good video despite that critique. I do think that with this kind of format, it is entertaining enough that you could make longer videos (e.g. with more concrete examples) and still keep people interested the whole way through. Just my 2 cents.

Cheers!

[tmplOS]

0:30-0:36 is absolutely hilarious to me for no reason.

[shadenineeighteen]

the way my Brain JUST RETAINED ALL THE INFORMATION 😮

[ANONAAAAAAAAA]

Riesz representation theorem is one of my favorites.

[pisadaseria]

What a fantastic strategy to discuss a topic; it makes the video very engaging. I wouldn’t be surprised if I ended up watching this for two hours.

[cdkw]

Damn the voice sound perfect now! Good work!

[KayKay-obtz]

U know what i like ur educative videos

[marsh_prootogn]

I love math, and their voice kinda calm me down

It's like a female version of organic chemistry teacher and prime newtons

[zru_]

WAIT the English channel gets them beforehand? 😳😳

[the_onion_peeler]

Ngl I don't understand most of this video but it's still entertaining

[wikichan]

0:37 looks like Metan doesn't like axiom of choice

[sunriseforg]

I’m damn interested into these videos, got my sub!

[뉴런-jh]

완전한 이해는 어렵지만 그럼에도 정말 흥미로운 내용이라는 생각이 드네요. 이런 영상을 만들어주셔서 감사해요.
Not being able to understand this video, I think this is SOOOO interesting. Thank you for your hard working