Recent content by Raven2816

Problem: u (sub t) = (1/2)u (sub xx) find the solution u(x,t) of the heat equation for the following initial conditions: u(x,0) = x u(x,0) = x^2 u(x,0) = sinx u(x,0) = 0 for x < 0 and 1 for x>=0 i'm really flying blind here. I've taken differential equations years ago but nothing...

then S2 is dense afterall, but not closed. ...at least from what I've worked out since.

ahhh i see what you mean! thanks!

a formula? isn't <Lx, Lx> = ||Lx||^2? and i know that L<x, y> = <Lx, y> ...

yes...e2 = e1 + e1 + 0?

Homework Statement i have {ej} is an orthonormal basis on a hilbert space S1 is the 1-dimensional space of e1 and S2 is the span of vectors ej + 2e(j+1) eventually i need to show that S1 + S2 is dense in H and also evaluate S2 for density and closedness Homework Equations i know...

and <Lx, Lx> is the inner product of an e-vector with its e-value...so do i get one? or am i using L=1?

because i know that the eigenvector is contained in a subspace spanned by sin(x) and cos(x). i figured that'd be a good f?

Homework Statement >M: L_2 -> L_2 > >(Mf)(t) = int(-pi, pi) sin(y-x)f(x) dx > >how do i find eigenvalues/vectors of M and what can i use to find >information about the spectrum? Homework Equations The Attempt at a Solution now i know that sin(y-x) = sinycosx-cosysinx...

heh, sorry about that. a unitary operator in a Hilbert space is what I'm working with

hmmm, i know that, so i have Ux = Lx and L is 1...so then Ux = x...? I'm just getting lost

Homework Statement why is the spectrum of the unitary operator the unit circle? Homework Equations i know that U^(-1)=U* and i know this makes U normal i also know that normal means UU*=U*U The Attempt at a Solution i know that from spectral theory there is some lambda in the...

so i can say that since that equality is true, and i can't sum uncountable positive terms, that obviously there are countably many a where the inner product is nonzero? is that a complete answer? it makes intuitive sense, but is it legit?

i know that <x, T_a> is non-zero for countably many a. I'm just struggling to show that this is in fact true.

i know there are countably many a such that for e>0 the absolute value of <x, T_a> is greater than e. i know that i can't sum an uncountable number of strictly positive terms because it doesn't converge. i just don't understand everything in the middle that should tie this together.