Functional Analysis Answers

show that the absolute value of a linear functional has the subadditive and positive homogeneous properties.

show that a norm on a vector space X is a sublinear functional on X

Q.1: Find the fourier series of f(x)=│x│ on [-π, π]
Q.2: Find the fourier series of f(x)=2-x2 on (-2<x<2)

1
limx→1(1+x+x2+……...+xm−1)
= …….
1
m-1
m
-1

if f:X to Y and g:Y to Z then the domain of gof is X and co-domain is Z , what is the domain and co-domain of fog, fof, gog, here in this case?

Let M be a sequence of the Hilbert space x . prove that M is dense in x iff ( Y⊥ M ⟹ y=0 )

the range of function, f(x)=x^3-1/x-1

A height of a shotput can be modelled by the function
H = -4.9t2 + 8t + 1.5
where H is the height in metres and t is the time in seconds.

1) At what point do you think the shotput was traveling the fastest? What factors did you use to make your inference?
2) Determine the average rate of change on a short interval near the point you chose in question 1.
3) Estimate the instantaneous rate of change at the point you chose in question 1
4) Were your answers to the average rate of change the same as the instantaneous rate of change, if not why not?

verify the equality: |u+v|2+|u−v|2=2|u|2+2|v|2, and. derive the theorem:the sum of the square so fthe diagonals of a parallelogram is equal to the sum of the square so the sides

What is the norm of the operator K from C[0, 1] into itself, if K is defined
by K (f )(s) = ∫ log |s − t|f (t) dt? (the integral taking from 0 to 1)