#1) This is incorrect, but you are not completely wrong. A linear functional on , i.e. an element of , is a function . You are right in saying that is defined on , but it is also possibly defined elsewhere. Therestrictionof to is an element of .

This is like taking the functional and projecting it on . Or, if you want to put it another way : is the image, under a (non-canonical) isomorphism , of some vector ; first projecting on and then sending, via , the resulting projection to , will give the restriction of to .

I haven't checked the details but I'm quite sure the above is right - you might want to check, I could just be high on coffee.