Ciro Santilli @cirosantilli 37

Convergence: math.stackexchange.com/questions/629630/simple-proof-euler-mascheroni-gamma-constant

Same value if you swap any input arguments.

The bioinformatics database: www.ncbi.nlm.nih.gov/

Here's a good example of what you can get out of it: E. Coli K-12 MG1655

Related: en.wikipedia.org/wiki/Protein_(nutrient)

In plain English: the space has no visible holes. If you start walking less and less on each step, you always converge to something that also falls in the space.

One notable example where completeness matters: Lebesgue integral of $L^p$ is complete but Riemann isn't.

When it exists, which is not for all matrices, only invertible matrix, the inverse is denoted:

Ahh, this dude is just like Ciro Santilli, trying to create the ultimate natural sciences encyclopedia!

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In 1995, Weisstein converted a Microsoft Word document of over 200 pages to hypertext format and uploaded it to his webspace at Caltech under the title Eric's Treasure Trove of Sciences.
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An impossible AI-complete dream!

It is impossible to understand speech, and take meaningful actions from it, if you don't understand what is being talked about.

And without doubt, "understanding what is being talked about" comes down to understanding (efficiently representing) the geometry of the 3D world with a time component.

Not from hearing sounds alone.

The dual space of a vector space $V$, sometimes denoted $V^{\ast }$, is the vector space of all linear forms over $V$ with the obvious addition and scalar multiplication operations defined.

Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the $V$, and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.

Infinite dimension seems more interesting however, see: en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case

One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space $V$.