Give an example of a positive operator on
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) Let with
. Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.
Give an example of a positive operator on
Let X be a Banach space and Y be a closed subspace of X. Let be canonical quotient map. Show that
is open.
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
Consider the space c00. For , define
. Show that f is a linear functional which is not continuous w.r.t the norm
.
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
b) C0 is a Banach space.
c) If A is the right shift operator on l2, then the eigen spectrum is non-empty.
d) If a normed linear space is reflexive, then so is its dual space.
e) If a normed linear space X is finite dimensional, then so is X'.
Consider the space C1[0,1] of all C1 functions on [0,1] endowed with the uniform norm induced from the space C[0,1], and consider the differential operator defined by
. Prove that D is linear, with closed graph, but not continuous. Can we conclude from here that C1[0,1] is not a Banach space? Justify your answer.