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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So it's now October 10, 2018 and....Rod Rosenstein is STILL not fired.
He's STILL in charge of the Mueller investigation.
He's STILL refusing to hand over the McCabe memos.
He's STILL holding up the declassification of the #SpyGate documents & their release to the public.
I love a good cover story.......
The guy had a face-to-face with El Grande Trumpo himself on Air Force One just 2 days ago. Inside just about the most secure SCIF in the world.
And Trump came out of AF1 and gave ol' Rod a big thumbs up!
And so we're right back to 'that dirty rat Rosenstein!' 2 days later.
At this point it's clear some members of Congress are either in on this and helping the cover story or they haven't got a clue and are out in the cold.
Note the conflicting stories about 'Rosenstein cancelled meeting with Congress on Oct 11!"
First, rumors surfaced of a scheduled meeting on Oct. 11 between Rosenstein & members of Congress, and Rosenstein just cancelled it.
He's STILL in charge of the Mueller investigation.
He's STILL refusing to hand over the McCabe memos.
He's STILL holding up the declassification of the #SpyGate documents & their release to the public.
I love a good cover story.......
The guy had a face-to-face with El Grande Trumpo himself on Air Force One just 2 days ago. Inside just about the most secure SCIF in the world.
And Trump came out of AF1 and gave ol' Rod a big thumbs up!
And so we're right back to 'that dirty rat Rosenstein!' 2 days later.
At this point it's clear some members of Congress are either in on this and helping the cover story or they haven't got a clue and are out in the cold.
Note the conflicting stories about 'Rosenstein cancelled meeting with Congress on Oct 11!"
First, rumors surfaced of a scheduled meeting on Oct. 11 between Rosenstein & members of Congress, and Rosenstein just cancelled it.
Rep. Andy Biggs and Rep. Matt Gaetz say DAG Rod Rosenstein cancelled an Oct. 11 appearance before the judiciary and oversight committees. They are now calling for a subpoena. pic.twitter.com/TknVHKjXtd
— Ivan Pentchoukov \U0001f1fa\U0001f1f8 (@IvanPentchoukov) October 10, 2018