Hello! I have got problems with the following exercise: Let be the continuous linear functional defined on the subspace in the Banach space with . Find different linear extensions on the whole space with same norm as !

I have calculated the norm of on , it is . My first idea was to extend to the basis and define .

But does this extension have the same norm? If yes, how can i show this? How do i find another extension? Hoping for your advice,

Recursion