The resolution of mathematics problems is easier when you have the right resources. But what are the right resources? The answer depends on who you are, what you are trying to achieve, and why.

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Ok, it's time for a #FUNctionalAnalysis thread! Let's talk about Hilbert spaces. (I hope you like linear algebra, because that's what we're
If you want to impress people, you can just say a Hilbert space is just a complete infinite dimensional inner product space and leave it at that, but let's talk about what that actually means.
When you first learn about vectors, you talk about them as arrows in space; things with a magnitude and a direction. These are elements of R^n where n is the number of dimensions of the space you care about.
You also talk about the dot product (or inner product) as a way to tell when vectors are orthogonal. (I'm purposely saying "orthogonal" instead of "perpendicular" here, but when you actually think about arrows, it's the same thing.)
As my linear algebra students are about to see, R^n is far from the only interesting vector space. A classic example is the space of polynomials of dimension less than or equal to n
— syzygay (@syzygay1) August 9, 2020
If you want to impress people, you can just say a Hilbert space is just a complete infinite dimensional inner product space and leave it at that, but let's talk about what that actually means.
When you first learn about vectors, you talk about them as arrows in space; things with a magnitude and a direction. These are elements of R^n where n is the number of dimensions of the space you care about.

You also talk about the dot product (or inner product) as a way to tell when vectors are orthogonal. (I'm purposely saying "orthogonal" instead of "perpendicular" here, but when you actually think about arrows, it's the same thing.)

As my linear algebra students are about to see, R^n is far from the only interesting vector space. A classic example is the space of polynomials of dimension less than or equal to n

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Once upon a time there was a Raja named Uttānapāda born of Svayambhuva Manu,1st man on earth.He had 2 beautiful wives - Suniti & Suruchi & two sons were born of them Dhruva & Uttama respectively.
#talesofkrishna https://t.co/E85MTPkF9W
Now Suniti was the daughter of a tribal chief while Suruchi was the daughter of a rich king. Hence Suruchi was always favored the most by Raja while Suniti was ignored. But while Suniti was gentle & kind hearted by nature Suruchi was venomous inside.
#KrishnaLeela
The story is of a time when ideally the eldest son of the king becomes the heir to the throne. Hence the sinhasan of the Raja belonged to Dhruva.This is why Suruchi who was the 2nd wife nourished poison in her heart for Dhruva as she knew her son will never get the throne.
One day when Dhruva was just 5 years old he went on to sit on his father's lap. Suruchi, the jealous queen, got enraged and shoved him away from Raja as she never wanted Raja to shower Dhruva with his fatherly affection.
Dhruva protested questioning his step mother "why can't i sit on my own father's lap?" A furious Suruchi berated him saying "only God can allow him that privilege. Go ask him"
Once upon a time there was a Raja named Uttānapāda born of Svayambhuva Manu,1st man on earth.He had 2 beautiful wives - Suniti & Suruchi & two sons were born of them Dhruva & Uttama respectively.
#talesofkrishna https://t.co/E85MTPkF9W

Prabhu says i reside in the heart of my bhakt.
— Right Singh (@rightwingchora) December 21, 2020
Guess the event. pic.twitter.com/yFUmbfe5KL
Now Suniti was the daughter of a tribal chief while Suruchi was the daughter of a rich king. Hence Suruchi was always favored the most by Raja while Suniti was ignored. But while Suniti was gentle & kind hearted by nature Suruchi was venomous inside.
#KrishnaLeela

The story is of a time when ideally the eldest son of the king becomes the heir to the throne. Hence the sinhasan of the Raja belonged to Dhruva.This is why Suruchi who was the 2nd wife nourished poison in her heart for Dhruva as she knew her son will never get the throne.

One day when Dhruva was just 5 years old he went on to sit on his father's lap. Suruchi, the jealous queen, got enraged and shoved him away from Raja as she never wanted Raja to shower Dhruva with his fatherly affection.

Dhruva protested questioning his step mother "why can't i sit on my own father's lap?" A furious Suruchi berated him saying "only God can allow him that privilege. Go ask him"
