What if I re-index so that n\in \mathbb{N} so that d_0=e_0, d_1=e_1, d_2=e_{-1},\dots. Then by Bessel's inequality we have \sum_{n=0}^{\infty}|(f,d_n)|^2\le \parallel f\parallel ^{2}<\infty. Hence \sum_{n=0}^{\infty}|(f,d_n)|^2 converges absolutely and whence (f,d_n)\to 0 and thus (f,e_n)\to\ 0...
Yep we have. Could I say that:
Since \{e_n\} is complete it follows that f=\lim_{n\to\infty} \sum_{m=-n}^{n}(f,e_m)e_m. Thus, the latter sum converges and hence \lim_{n\to\infty}\sum_{m=-n}^{n}|(f,e_m)|^{2}<\infty. Thus |(f,e_m)|^2 \to 0 so |(f,e_m)|\to 0 and whence (f,e_m)\to 0, as required...
Wait, if {e_n} is complete, then \lim_{n\to\infty}\sum_{m=-n}^{n}(f,e_m)e_m converges to f absolutely, so the coefficients necessarily converge to zero. Would this work?
Homework Statement
Let e_{n}(t)= \frac{1}{ \sqrt{2\pi}}\cdot e^{int} for n\in\mathbb{Z} and -\pi\le t\le\pi.
Show that for any f\in L^{2}[-\pi,\pi] we have that (f,e_{n})=\int_{-\pi}^{\pi}f(t)\cdot e^{-int}dt\to0 as |n|\to \infty.
The Attempt at a Solution
I want to use dominant convergence...
I know that thinking a R-module is simply a ring R acting on a set (following the usual axioms), would it be safe to think of an R-algebra as the ring R acting on another ring?
This may seem convoluted, but I'm just having a little trouble getting through all the different definitions of an...
Well it is trivial if m=0, so suppose m\neq 0 even. Then it follows that m=2k hence m \otimes 1 = 2k\otimes 1 = 2(k\otimes 1)
Hence, for any morphism of \mathbb{Z}-modules \phi : (\mathbb{Z}_{10}\otimes_{\mathbb{Z}}\mathbb{Z}_{12})\to \mathbb{Z}_{2}, it follows that...
Homework Statement
Show that \mathbb{Z}_{10}\otimes_{\mathbb{Z}}\mathbb{Z}_{12} \cong \mathbb{Z}_{2}
The Attempt at a Solution
Clearly, for any 0\neq m\in\mathbb{Z}_{10} and 0\neq n \in \mathbb{Z}_{12} we have that m\otimes n = mn(1\otimes 1), and if either m=0 or n=0 we have that m\otimes n =...
Homework Statement
Let H be a Hilbert space. Prove \Vert x \Vert = \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert}
The Attempt at a Solution
First suppose x = 0. Then we have \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert (0,y)...
Algebraic Topology
Algebraic Geometry I
Linear Algebraic Groups
Functional Analysis II
Rings and Modules
Field Theory
Commutative Algebra
Lie Algebras
Differential Equations
Algebraic Geometry II
I bet you wouldn't guess that I'm applying to an algebra group for graduate studies...