📝UPDATE: As I mentioned earlier in the comments this proof seems far from convincing as in the last equation we can not easily cancel two terms which both are absolute zero. We can say there is no absolute solution for the 0 ^ 0. Some think we should define it as 1 and others as undefined. I agree with the latter more. However, defining it as 1 brings convenience in some math problems. Thus, for more information feel free to search and read and if you find interesting one share with me.

This subject is also discussed on a Wikipedia page.
@SingularThinker

"ریاضیات چیزی نیست جز بیان چیزهای یکسان به روش‌های مختلف."

این جمله که نقل قولی از یک ریاضیدان هست رو از بخش آخر بلاگ امیر اصغری که در مورد جادوی جدول ضرب بود، آوردم. هم جمله جالبیه هم بلاگش، اگه دوست داشتید ببینیدش.

https://math.omidedu.org/magic-of-multiplication-table/

#math
@SingularThinker

What is a Hilbert space, really?

It is just a fancy name for a Cauchy complete inner product vector space. 

Wait what? I'll explain.

First, why do we need inner product vector space? 

Inner product vector spaces have physical meaning and usefulness more than any arbitrarily vector space. 

Now lets consider combining many stones together to understand why Cauchy's completeness is necessary. You expect this to result in a stone object, don't you? I should say that it depends on how many stones you combine. If there are infinitely many ones, it may change the situation. For a more visual interpretation, I suggest watching this video (the initial minutes can be skipped).


In the end, why are Hilbert spaces useful?

We can construct orthogonal sets via Hilbert spaces.

🔗 Reference
#video #math

P.S: Also in some references said that a Hilbert space should be separable with respect to the norm defined by the inner product.
@SingularThinker

...

How to make a Hilbert space from scratch? | Part 1/2

🔗 Link

#note #math
@SingularThinker