What is the definition of a k-chain of polyhedrons in a Banach space? What do you mean by a 'sum' of polyhedrons?
Right, we'll go through this one step at a time:
A Banach space is a complete normed vector space. That is: start with a field (probably
R or
C), and a set. Define two functions: one ("scalar multiplication") that takes an element of your set (we'll call them vectors) and an element of your field (we'll call them scalars) and gives you another vector, and another ("addition") that takes two vectors and gives you a third, and have them work exactly as you would expect addition of vectors and scalar multiplication to work in
Rn (including having a vector (that we'll call 0) such that 0 + x = x for all vectors x - you can think of this exactly as you would n-dimensional space, except that it may be infinite dimensional, which makes weird stuff happen. Now define a third function (our "norm" - we'll denote the norm of a vector x by |x|) that takes vectors and gives you a non-negative real number that's supposed to represent how "big" that vector is (or "how far it is from zero", if that helps). Have this function be something sensible as far as representing a distance from zero goes (only zero on the zero vector itself, never negative, multiplying the vector by a scalar multiplies the norm by the absolute value of that scalar, and the norm of the sum of two vectors is no bigger than the sum of the norms - if it seems a bit weird that it's an inequality, think about adding the vectors (1,0) and (0,1) in
R2, and notice that the line from 0 to (1,1) is shorter than 2.
That brings us to completeness. This one's a little harder to explain, but roughly, it means that there's no gaps in your space (relative to your norm) - for example, the rational numbers aren't complete, they're full of little gaps (the irrational numbers), but the real numbers are (no, I'm not going to prove that here). To be more precise: If we take any infinite sequence of vectors in our space, say it is Cauchy if they are "eventually arbitarily close together" - that is, a sequence (a_n) is Cauchy if, for every e > 0, there is a N such that for all n and m > N,
|a_n - a_m| < e. Say that the sequence converges if there is a vector a in our space such that the vectors are "eventually arbitarily close to a" - that is, (a_n) converges to a if, for every e > 0, there is a N such that for all n > N, |a_n - a| < e. If you think about this for a while, you might think that every Cauchy sequence should converge - and this is precisely what it means for a space to be complete. You can think of Cauchy sequences that don't converge as sequences that are trying to converge, but the thing that they're trying to converge to is missing, roughly (this is where the "no gaps" thing comes from).
Now, if all of that seemed horrible, here's the good news: in finite dimensions, they're all basically the same as
Rn with the standard "square the coordinates, add them up and square root it" measure of size that you learnt in school.
I presume you know what "polyhedron" means in normal, 3 dimensional space. Here, we're talking about something that's basically the same, but in a more general banach space: the convex hull of finitely many points in a 3-dimensional subspace of that Banach space (or probably any embedded copy of
R3) - what this means is that you take anything in your plane that's basically 3 dimensional (think of taking a flat slice of 3-dimensional space, then try to think about taking a 3d slice of 4d space, then just stop worrying about how many dimensions the big space has), pick some (finitely many) points in it, join them together with lines, and fill in the inside of the shape that you get. There's probably some non-degeneracy condition as well (no picking all of the points in a straight line/plane, probably). He might also be using the term to mean polytopes in general (it makes more sense with the k-chains later); which is exactly the same, except rather than only picking 3d slices, we pick slices of any (finite) number of dimensions n out of our space, then proceed as above (with the non-degeneracy condition (if it exists) now probably being something like "not all lying in the same n-1 dimensional space).
Now, on to sums of things that don't make sense to have sums: This is one of the single least satisfying definitions in all of mathematics, but the formal sum of some set of objects literally just means "write them down with + signs in between (and coefficients on the front so you can write 3x instead of x+x+x)". Whilst there are ways to formalise it with free abelian groups and stuff, basically nobody ever bothers, and that's what it comes down to. You can add things up exactly as you'd expect
A k-chain of polyhedron is now easy to define: it's a formal sum of k-dimensional polytopes (this is why I think he's talking about polytopes in general, rather than 3 dimensional ones in particular - if I were talking about the latter, I'd miss the "k" off the start). You can then define some actual content on these things: you can form something called a group (somewhere you can add up [as above], have something that acts like zero when adding things [the empty chain: the formal sum of no things], and you have a "-x" for every x [the same chain with all coefficients replaced by minus themselves] - all of this is true of general formal sums, and is exactly that formalisation that I was talking about above), and you have a boundary operator, that takes a k-chain of polyhedra, and gives the chain of (k-1 dimensional) polyhedra made up of its faces (so it would take a cube to the sum of the six squares that make up its faces, for example).
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Topic: Future of the Fortress (Read 1884116 times)