Inner product in normed space

Question

Suppose $(X, \| \|)$ is a normed space and an inner product $<,>$ is defined on $X$ such that the induced norm $\| \|_i$ is different from the original norm. Is it true that there must exist a constant $a$ such that $|<x,y>|\leq a \|x\|\|y\|$?

Answer 1

No. Let, for instance, $X$ be the space of all sequences $(x_n)_{n\in\mathbb N}$ of real numbers such that $x_n=0$ if $n\gg1$. Define$$\bigl\lVert(x_n)_{n\in\mathbb N}\bigr\rVert=\sup_{n\in\mathbb N}\lvert x_n\rvert.$$Now, let$$\bigl\langle(x_n)_{n\in\mathbb N},(y_n)_{n\in\mathbb N}\bigr\rangle=\sum_{n=1}^\infty x_ny_n.$$If, for each $N\in\mathbb N$, $x(N)\in X$ is the sequence$$\overbrace{1,1,1,\ldots,1}^{\text{first $N$ terms}},0,0,0,\ldots,$$then$$(\forall N\in\mathbb N):\bigl\lVert x(N)\bigr\rVert=1,$$but$$(\forall N\in\mathbb N):\bigl\langle x(N),x(N)\bigr\rangle=N.$$And you can't have$$(\forall N\in\mathbb N):N\leqslant a.$$