Exercise:
Let $(X,\Vert \cdot \Vert)$ be a normed $\mathbb R$-vector space. Which of the following statements are true and which are wrong?
1.) For each $x \in X$ there exists $f \in X^*$ (dual space) with $\Vert f \Vert \le 1$ so that $\Vert x \Vert = f(x)$
2.) For each $f \in X^*$ there exists $x \in X$ with $\Vert x \Vert \le 1$ so that $f(x) = \Vert f \Vert$
3.) $X$ is finite dimensional if and only if every linear functional is continuous.
4.) If $X$ is finite dimensional, then each hyperplane in $X$ is closed.
My Questions:
For 1.) and 2.) I don't know where to start. It would be great if someone could give me a hint.
For 3.) I know that if X is finite dimensional then every linear functional is continuous. But the other implication does not have to be true. Right?
For 4.) As I know that X is finite dimensional this implies that every linear functional is continuous. There is a proposition which says that if f is continuous than the hyperplane $H=f^{-1}(\{\alpha\})$ is closed. So this should be enough to argue that the statement is true?
Thanks in advance for your help!