Functional Analysis Answers

If there are 20 items on his list on the 26thof March, and the number of items grows by

25% per day, how many items will he have on his list on April 9th?

Find all metrics on a set X consisting of two points. Consisting of one point only.

Let the functional f on R² be defined by f(x)= 4x-3y.Regard R² as a subspace of R³ given by z=0 determine all linear extension of f(x) from R² to R³

• Prove that C.V100√n-1

Let V be a Banach space and let M ⊆ V be a proper subspace of V (i.e. M not equal to V ). Prove that if v ∈ V and v /∈ M then there is a φ ∈ V* such that φ(v) = 1 and φ(w) = 0 for every w ∈ M.

(i) Let V be a Banach space. Prove that if V* separable then V is not separable.

(ii) Give an example of separable Banach space V which has a non-separable dual space V* . 

Let c0 be the space of sequences of complex numbers which converge to 0. That is

c_0 = {(x_i)_i∈N : xi ∈ C, x_i → 0}.

(i) Show that c0 is a closed subspace of L^∞.

(ii) Define a mapping T by

T : L^∞ → c_0

(x_n)_n → (x_n/ n ) _n

Show that T is a (linear bounded) operator. Show that ran T is not closed.

• Prove that C.V100√n-1

Define f(x) = sinx on [0, 2pi]. Find two increasing functions h and g for which f = h — g on 

[0, 2pi]. 

Show that f(x)=2/x+1 is indected