Good one. If I remember correctly, once you get up to working in function spaces it's time for the Minkowski inequality for many of the same applications that the triangle inequality enables.
Yes, it is. Non-Euclidean geometries take place in metric spaces, and the triangle inequality is required of a metric. The triangle inequality is foundational for the notion of "distance" or "length" to make sense. Its essence is that the distance from a to b reports the length of the shortest possible route from a to b. That's what we want distance to mean. We don't want a to be 15 miles away from b, but somehow if I go from a, to c, and then to b, I ended up traveling only 12 miles in total.
I was hoping you could prove the Reverse Triangle Inequality. I'm also wondering why we need to establish |a-b| is greater than the difference of their absolute values...
My linear algebra professor used to call it the donkey inequality because it’s so simple, even a donkey gets it! (The donkey of course will walk a straight line to get to water, not go in a jagged path.)
It is true coz if both a and b are positive then abs(a+b)=abs(a)+abs(b) but if one of then is negative; let’s say b is negative and let’s say that a=32 and b is equal to -31, then abs(a+b)=abs(32-31)=1; but it is not equal to abs(32)+abs(-31)=32+31=63;in fact the abs(a)+abs(b) is greater than or equal to abs(a+b). :))))
I remember my Math Professor saying..." Learn as much as you can about inequalities, you will need them very very often"
More often than not. :P
If you're in first year of university or last year of school then take this advice seriously . From a panic cramming 2nd year maths student
That triforce in the thumbnail made me click as fast as I could
I wasn't expecting a dab in a math video. Excellent!
Good one. If I remember correctly, once you get up to working in function spaces it's time for the Minkowski inequality for many of the same applications that the triangle inequality enables.
what a coincidence that i have to prove this for my teacher and you posted this 1 day ago🤩🤩 thank uu
The "dab" you have to train... xD
8:30 i did not understand a word here , please if some one could reexplain it to me
Is this triangle inequality also true for triangles in non euclidean spaces?
Yes, it is. Non-Euclidean geometries take place in metric spaces, and the triangle inequality is required of a metric.
The triangle inequality is foundational for the notion of "distance" or "length" to make sense. Its essence is that the distance from a to b reports the length of the shortest possible route from a to b. That's what we want distance to mean. We don't want a to be 15 miles away from b, but somehow if I go from a, to c, and then to b, I ended up traveling only 12 miles in total.
Did he just do the dab 9:10😂😂
I was hoping you could prove the Reverse Triangle Inequality. I'm also wondering why we need to establish |a-b| is greater than the difference of their absolute values...
There’s a video on that
The Cauchy Schwarz inequality is driving me mad! Will you do a video on that too?
Already done ✅
@@drpeyam Whoops! Thank you Peyam :D
but actually it's Cauchy-Buniakovsky inequality
I'm wondering why you don't consider the proof by using the dot product (a+bIa+b), seeing as the proof comes out of it naturally.
That’s the way I would have done it as well, but in that class they don’t really mention dot products or R^n
@@drpeyam Oh cool.
Sir , but is there any example or case where one side will be equal to to sum of other side in a triangle ? The inequality says lesser or equal to
when the triangle is degenerate(smushed down into a line) this is true
11:34 *"the legthgh"*
I'm looking forward to Jensen's Inequality. :-)
That’s a good one 😄
My linear algebra professor used to call it the donkey inequality because it’s so simple, even a donkey gets it! (The donkey of course will walk a straight line to get to water, not go in a jagged path.)
LOL, I love this 😂
Dr Peyam he still teaches at CSUF, though I guess you’re leaving UCI in the fall.
Awesome explanation!
Why in proof 1. case 1 when x ≥ 0 then x = |x| ≤ -|x| ; and not
x = |x| ≥ -|x| ?
It’s a typo I think
@@drpeyam I see. Thankss
was that the triforce??
Yeah
@@drpeyam pogggggg
It is true coz if both a and b are positive then abs(a+b)=abs(a)+abs(b) but if one of then is negative; let’s say b is negative and let’s say that a=32 and b is equal to -31, then abs(a+b)=abs(32-31)=1; but it is not equal to abs(32)+abs(-31)=32+31=63;in fact the abs(a)+abs(b) is greater than or equal to abs(a+b). :))))
Great video! Thanks a lot for this :)
Lmao I love the dab!!!! 🤣🤣🤣
You make it more harder for me😢🙆🏽♀️
Wait what? 🤯🤯, I.. I'm new to this
you could define abs(x) = Max (x,-x), then your lemma is evident
dam you're fast answerer, hello from France!
I trust my Toyota Corollary.
Nice, thanks
the triforce :skull zelda refrence
good video!
subscribed just after the dab ; )
Hi dr 💚
Analysis is adult version of calculus. Isn't it??
It really is!
Next measure theory
i prefer algebra.
Professor ! Why It is called Triangle Inequality ?
Look at the picture
👍
The explanation's a bit hard to grasp Doc. Its a bit too complex
Good for you
@@drpeyam i dont think its good for him that he didn't grasp it :(