Functional Analysis Answers

Suppose the function f is continuous on the interval [a, b] and never zero on [a, b].
Is it possible that f(z) < 0 for some z ∈ [a, b] and f(w) > 0 for some w ∈ [a, b]?
Explain your answer.

Suppose f is a function such that
f(x) = x^2 x element Q,
0 x element R − Q.
Is f continuous at c = 0? Is f differentiable at c = 0?

prove that two norm linear spaces may be topological isomorphic but not necessary congruent give example.

514 - (372 - 547 x (330 - 453)) = ?

consider a cubic function f:x-ax^3+b,where a and b are real numbers and a not equal to 0.the shape of the cubic function depends on the value of a and b.
a) By using any sitable tools.investigate the shape of the graph if both a and b have the same sign and if a and b have different sign. identify the point of inflexion in eash case.
b) Investigate the point of intersection of the graph of f and its tangent.What can you say about the number of point of intersection?

Let X be complex Banach space , T Є B(X,X) and p a polynomial .Show that the equation p(T)x = y has a unique solution x for every y ЄX if and only if p(λ)≠0 , for all λ Є σ(T)

Show that K(x-a) = O(k-a) [this is the letter O no the number 0] for any K.

A subspace Y of a normed space X is said to be invariant under a linear operator T:X→X if T(y) ЄY , Let λЄσp(T) (λ belongs to point spectrum), TεB(X,X) , X be a complex Banach space. Show that eigen space of λ is T-invariant.

Let TεB(X,X) ,X is a comlex Banach space and T be idempotent(i.e T^2=T) .Prove that if T is neither 0 not I ,then σ(T)={0,1}

Let TεB(X,X) be invertible in B(X , X) , X is a complex Banach space.Show that σ(Inverse of T)={1/λ : λЄσ(T)}