Homework Statement
So this is arising in my applied math course in solving the wave equation in n dimensions. So we have a function u(\vec{x}+r\vec{z},t) and where x and z are n dimensional vectors and r is a scalar (also, u is a scalar function). Then when we take the partial derivative with...
Homework Statement
My linear algebra is rusty. So to go from a reduced QR factorization to a complete QR factorization (ie the factorization of an over determined matrix) one has to add m-n additional orthogonal vectors to Q. I am unsure on how to find these.
If it is extending a 3x2 to a 3x3...
Homework Statement
So the context is this arises in the method of descent, for finding a solution for the 2D heat equation from the 3d heat equation. Anyway, in one step, we must change the surface integral over a ball in 3d, to the surface integral over it's projection into a plane. In this...
Haha, sorry. So this all comes up in a proof for "the closed convex hull of a compact subset of a completely metrizable space is compact". It can be found on page 186 here...
So if they weren't unbounded, then you could cover it with a ball of radius bigger than the largest element? I think I am okay with this, but in the proof where this was stated, it was for an convex neighborhood about 0 which is a subset of any arbitrary open neighborhood of 0. So in turn it...
Homework Statement
So I am trying to understand this proof and at one point they state that an arbitrary compact subset of a Banach space, or a completely metrizable space is the subset of a finite set and an arbitrary convex neighborhood of 0. I've been looking around and can't find anything...
Hmm, can't all points in the unit sphere in say 3d be written as linear combinations of (1,0,0), (0,1,0) and (0,0,1)? Is your point about semantics or is it really erroneous to carry over these concepts to a general space? I thought that linear spaces have the same... I don't what to call it...
So would it be correct to say that a compact subset of a normed linear space (B* or Banach) must have a finite basis or has a finite dimension?
Thanks a lot!
Homework Statement
Hi all, I am struggling with getting an intuitive understanding of linear normed spaces, particularly of the infinite variety. In turn, I then am having trouble with compactness. To try and get specific I have two questions.
Question 1
In a linear normed vector space, is...
One thing is that ||y-x|| > C for a fixed y and any x and I think that in the counter example you gave for any fixed y, the infimum of ||y-x|| would be equal to ||y-r|| where r is actually in X. Does the y being fixed change anything?
Homework Statement
So this question arose out of a question about showing that a set χ is dense in γ a B* space with norm ||.||, but I think I can safely jump to where my question arises. I think I was able to solve the problem in another way, but one approach I tried came to this crux and I...
Homework Statement
Hello, we are starting to get to Banach spaces and thus linear normed spaces in a functional analysis class and I am realizing that I don't have much experience or intuition with these spaces. So I was reading over the requirements for a linear space in my notes and was...