and while it's not something I know a lot about, i know that in spectral theory they care a lot about objects of the form $(z I-A)^{-1}$ where $A$ is some matrix and $I$ an appropriate identity matrix.
in that case you're no longer working with polynomials, but you still get useful results regarding how to decompose that matrix inverse into a sum of other inverses.
It occurs to me they might also come into play in perturbation theory, where you might want to reduce a 2nd-order approximation in a quotient to the sum of two first-order approximations, which might "blow up" slower
I am reading the book Perturbation theory for linear operators from Kato.
He defines (ยง5 Section 3) for an operator $T : X\to X$ on a finite Banach Space the resolvent as
$$ R(x) = (T- x)^{-1}.$$
He then consideres the Laurent Series of the resolven on a singularity $\lambda$ of $R$ and find...
In the complex plane, for example, it's simpler to go around holes, "one at a time", then have to worry about a region "peppered" with them. One hole is easy to miss.
for the former, maybe you could characterize the sequence as being analogous to an infinite column vector, and so the dual would be the space of infinite row vectors
Those two are just examples, of course. In general I somehow don't understand the duals of infinite-dimensional spaces, which seem to be much more complicated than their finite-dimensional cousins
HI I'm working with Conway's analysis book and in one question I really appreciate a clarification. This says: Find an open connected set G in the complex plane and two continuous functions f and g defined on G. s.t. f^2= g^2 = 1-z^2 for all z\in G. Can you make G maximal? Are f and g analytic?
My approach is the following, since when we taking a branch for the square root we cut some ray emanating from the origin we can choose for \sqrt (1-z) and \sqrt (1+z) two rays say (-\infty, -1] and [1,\infty) and define G as the complex plane minus the two rays.
as can be seen from the fact that the following is a well-defined mapping to the reals: "What is the sum of all the terms of my sequence (including all the ones that are zero)?"
i.e. the only way to make sure you get a finite result from an arbitrary sequence in $\mathbb{R}^\mathbb{N}$ is to take only finitely many of its terms
But of course, the set of functions nonzero at a finite number of points is a strict subset of R^R, so there must be other classes of linear functionals that I've missed
hmm. from wikipedia's page on schauder bases: "In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces."
With a Hilbert space (a "normal" kind of infinite-dimensional vector space), if we restrict our attention to continuous linear functionals, things "get nice" again.
hey everyone! I have an interesting question (from a calc course). Basically I'm dealing with an "infinite trumpet" (i.e. $1/x$ revolved around the x-axis where $x \ge 1$). The question has asked me to find volume and surface area. I did this. Then it poses the question: Suppose we fill up the trumpet with the finite amount of paint (equal to the volume), it would seem that we have coated the infinite interior SA with a finite amount of paint. How is this possible?
@DavidWheeler for context, i'm starting with a $n$-variable function $f_n(t_1,t_2,\ldots,t_n)$ and wanting to express $f_{k-1}(t_1,\ldots,t_{k-1})$ and $f_{n-k}(t_{k+1},\ldots,t_{n})$ as simply as possible
Oh my god yall fuckin' haters need to S T O P STOP! I'm tryin to get smart here and yall fuckin turbonerds stoppin me from achievin' my dreams! WTF? — xXxSkIttLEzDeStRoYeRxXx2 mins ago
I need yo help. Look see, mah girl was like "You don't love me Romeo" and I'm like "Baby I love you" and shes like "No you font you made me lobster and then didn't turn on the water chef boy-r-dumb" and I'm like "how can i prove that I love you?" and shes like "Get a brain."
How do i do some of ...
@Semiclassical there was an interesting discussion by Stan Shunpike about the idea of "distance" in the physics room where he quotes what people said to him in here.
@DavidZ I was talking to some dudes in the math chat and they said something to the effect that...distance is a useful way to get spatial information just not necessarily information that can be described with real numbers. But thats math. Agree or disagree when it comes to physics?
heh, now this is a fun formula which i don't know how to proceed with further:
for a particular n-variable function $F_n$, I found $$dF_n(x_1,x_2,\ldots,x_n)=\sum_{k=1}^n F_{k-1}(x_1,\ldots x_{k-1}) F_{n-k}(x_{k+1},\ldots x_{n-k})\,dx_k$$
I see this image just about every time I see someone talk about the geometry of space. I was generally following the lecturer's analogies and descriptions in my mind pretty well, and liked the idea of using a the sum of angles in a triangle to bring the point home for different values of omega. Actually looking at the triangle when omega < 1 always threw me off though. I see another triangle with angles > 180 degrees.
I am reading the book Perturbation theory for linear operators from Kato.
He defines (ยง5 Section 3) for an operator $T : X\to X$ on a finite Banach Space the resolvent as
$$ R(x) = (T- x)^{-1}.$$
He then consideres the Laurent Series of the resolven on a singularity $\lambda$ of $R$ and find...
The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind:
$$U_n(x)=\begin{vmatrix}2 x& 1 & 0 &\cdots &0\\ 1 & 2x &1 &\cdots &0 \\
0 & 1 & 2x &\cdots &0\\0 & 0 & 1 & \ddots & \vdots \\
\vdots & \ddots & \ddots &\ddots &
