Answer to Question #206300 in Functional Analysis for Ibtehal ahmed

Question

Let p be defined on Vector Space X and satisfies p(x+y)≤p(x)+p(y) and for every scalar a, p(ax)=|a|p(x). Show that for any given x′∈X there exists a linear functional f′ on X such that f′(x′)=p(x′) and |f′(x)|≤p(x) for all x∈X.

Answer 1

Linear transformation:

a function T : V → W such that for all α, β ∈ F and x, y ∈ V , T(αx + βy) = αT(x) + βT(y)

Let X be a normed space. Linear transformations from X to F are called linear functionals.

Then:

p(x'+y')≤p(x')+p(y') - we have.

Since f'(αx + βy) = αf'(x) + βf'(y) then

f'(x'+y')=f'(x')+f'(y') or f'(x'+y')$\le$ f'(x')+f'(y')

and, also:

let x=ax' then:

p(x)=p(ax')=|a|p(x') - we have.

|f'(x)|=|f'(ax')|=|af'(x')|

So, if f′(x′)=p(x′):

|f'(x)|=p(x) or |f'(x)|$\le$ p(x)