Prove that $Y^* \subset X^*$

Question

I don't know for what I had to begin - prove that if X and Y are total space and $X \subset Y $ than $Y^* \subset X^*$, where X*, Y* - dual space.

Answer 1

If $f$ is a bounded linear functional on $Y$ then $f$ is certainly a bounded linear functional on $X$ (which is a subset of $Y$).

The idea is that it's easier to be bounded on $X$, which is the smaller space, than it is to be bounded on $Y$, which is the whole thing.