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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Stan Lee, who died Monday at 95, was born in Manhattan and graduated from DeWitt Clinton High School in the Bronx. His pulp-fiction heroes have come to define much of popular culture in the early 21st century.
Tying Marvel’s stable of pulp-fiction heroes to a real place — New York — served a counterbalance to the sometimes gravity-challenged action and the improbability of the stories. That was just what Stan Lee wanted. https://t.co/rDosqzpP8i
The New York universe hooked readers. And the artists drew what they were familiar with, which made the Marvel universe authentic-looking, down to the water towers atop many of the buildings. https://t.co/rDosqzpP8i
The Avengers Mansion was a Beaux-Arts palace. Fans know it as 890 Fifth Avenue. The Frick Collection, which now occupies the place, uses the address of the front door: 1 East 70th Street.
