# Norm on normed algebra

I am reading a book " Abstract Linear Algebra " - M. L. Curtis

Chapter V is about normed algebras, which is a title. Here normed algebra $X$ is a algebra with norm satisfying the following relation : For $a$, $b\in X$, $|ab|=|a||b|$.

${\bf R}$ is a typical example. So from this we want to consturct ${\bf C}$. So we must define multiplication structure. The book insist that since the norm is comed out from inner product, the multiplication is unique.

From the parallelogram law we can derive inner product from norm. If not I do not understand why the title is normed algebra.

Hurwitz theorem is that the only real normed algebras are ${\bf R}$, ${\bf C}$, ${\bf H}$, and ${\bf O}$.

I will summarize my question : Can we derive inner product from normed algebra ?

My answering : Let $(s,t)=(0,1)^2$ (i.e., $s^2 + t^2=1)$. If $(1,0)$ is a left identity for multiplication, then $2=|(1,1)(0,1)|^2= |(0,1) + (s,t)|^2=2+2t$ Hence $t=0$.

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First, take a look at this MO thread on the same topic. Notice that the accepted answer cites that any normed division algebra is finite-dimensional and therefore isomorphic to the four Hurwitz's theorem describes $\Bbb{R}, \Bbb{C},\Bbb{H}, \text{and} \,\Bbb{O}$. Therefore your result should hold, as all of the norms on those spaces fulfil the parallelogram law.