(a) Define Banach spaces and Euclidean spaces over $\mathbb{R}$. [You may assume the definitions of vector spaces and inner products.]

(b) Let $X$ be the space of sequences of real numbers with finitely many non-zero entries. Does there exist a norm $\|\cdot\|$ on $X$ such that $(X,\|\cdot\|)$ is a Banach space? Does there exist a norm such that $(X,\|\cdot\|)$ is Euclidean? Justify your answers.

(c) Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{R}$ satisfying the parallelogram law

$$\|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}$$

for all $x, y \in X$. Show that $\langle x, y\rangle=\frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)$ is an inner product on $X$. [You may use without proof the fact that the vector space operations $+$ and are continuous with respect to $\|\cdot\|$. To verify the identity $\langle a+b, c\rangle=\langle a, c\rangle+\langle b, c\rangle$, you may find it helpful to consider the parallelogram law for the pairs $(a+c, b),(b+c, a),(a-c, b)$ and $(b-c, a) .]$

(d) Let $\left(X,\|\cdot\|_{X}\right)$ be an incomplete normed vector space over $\mathbb{R}$ which is not a Euclidean space, and let $\left(X^{*},\|\cdot\|_{X^{*}}\right)$ be its dual space with the dual norm. Is $\left(X^{*},\|\cdot\|_{X^{*}}\right)$ a Banach space? Is it a Euclidean space? Justify your answers.