Homework Statement
Let f,g be two continuous, periodic functions bounded by
[-\pi,\pi]
Define the convolution of f and g by
(f*g)(u)=(\frac{-1}{2\pi})\int_{-\pi}^{\pi}f(t)g(t-u)dt.
Show that
(f*g)(u)=(g*f)(u)
The Attempt at a Solution
I think the way I'm...
Homework Statement
Let V be a finite dimensional normed vector space and let U= L(V)*, the set of invertible elements in L(V). Show, f:U-->U defined by f(T)= T-1 is differentiable at each T in U and moreover,
Df(T)H = -T-1HT-1
where Df(T)= f'(T).
Homework Equations
Apparently...
So, for the first property, I know that ker(T)={0} so then
||x||' = ||Tx|| = 0 iff x=0, but how do you know that ||Tx||>=0? Is it just because ||x|| is a norm?
For the second, I know
||cx||' = ||T(cx)|| = ||cT(x)|| = |c|*||Tx|| = |c|*||x||', so this holds.
For the...
Homework Statement
Suppose T : V --> W is a linear transformation and one-to-one. Show, if ||.|| is a norm on W, then ||x|| =||T(x)|| is a norm on V.
(V and W are vector spaces)
Homework Equations
T is linear, so T(x+y)= T(x) + T(y) and T(ax)= aT(x)
T is one-to-one, so T(x)=T(y)...