Question 1. Let Xand Ybe two normed linear spaces and let T:X→Ybe a bounded linear
operator from Xto Y.
a. Define the norm of the operator T.
b. Find the norm of the operator T:Lp(0,2) →Lp(0,2),1≤p < ∞, where
T f (t) = t2f(t), t ∈(0,2).
Question 2. Let Xand Ybe metric linear vector spaces and let T:X→Ybe a linear operator
from Xto Y.
a. Define the graph of the operator T.
b. Give definition of a closed operator.
c. Show that the operator Tin L2(0,1) with D(T) = C[0,1], such that
T f =f(0),
is not closable.
Question 3.
a. What is the Schwartz inequality in a Hilbert space? Prove it.
b. What is the Schwartz inequality for the Hilbert space L2(R).
c. What is Bessel’s inequality? Prove it.
Question 4. Let H1(0,2π)⊂L2(0,2π)be a Hilbert space with the scalar product given by
(f, g ) = Z2π
0f0(x)g0(x) + f(x)g(x)dx
and let
en(x) = aneinx.
a. Show that {en}n∈Zis an orthogonal system in H1(0,2π). Find the coefficients an, such that it
is an orthonormal system.
b. Find a function f∈H1(0,2π)such that fis orthogonal to all en,n∈Z, in H1(0,2π).
