Problem with the Definition of Inner Product Space.

Question

Let $V$ be a vector space over a field $F=\mathbb{R}/\mathbb{C}.$ Then an inner product on V is defined to be a function $(,):V\times V \rightarrow \mathbb{F}$ satisfying the following property:

Linearity in the first coordinate.
Conjugate-Symmetry.
Positivity: For all $x \in V$, if $x$ is nonzero then $(x,x)>0$.

My Questions:

If I take $\mathbb{F}$ to be $\mathbb{C}$. Since it is not an ordered field, the idea of positivity does not make sense. But I find this definition everywhere. Am I missing something out?
Every Inner Product defined on $V$ (an $\mathbb{R}$-vector space) is bilinear. Does every bilinear map $f: V\times V\rightarrow \mathbb{R}$ give rise to an inner product on $V$?

Because of Property 2 one gets that $(x,x)\in \mathbb{R}$. – mfl Dec 21 '20 at 20:12 — mfl, Dec 21 '20 at 20:12

José Carlos Santos · Accepted Answer · 2020-12-21T20:47:08.770

1

The third condition should be: for every $x\in V$, $(x,x)\in[0,\infty)$. In particular, $(x,x)$ is always a real number.
No. If, say $f(x,x)$ is always equal to $0$; then $f$ is not an inner product. And, if $V=\Bbb R$ and $f(x,y)=-xy$, this $f$ is also not an inner product.

edited Dec 21 '20 at 20:47

answered Dec 21 '20 at 20:11

José Carlos Santos

440,053

Thanks for the answer. I have added one more question to this. Can you please provide an answer to the added question? – Saikat Dec 21 '20 at 20:44
@Saikat Done.${}$ – José Carlos Santos Dec 21 '20 at 20:47
By $f(x,x)=0$ always, you mean a bilinear map with this property. Correct? That is if we take $f(x,y)=0$ always. Then this works well. Am I correct? – Saikat Dec 21 '20 at 20:57
Conjugate-symmetry tells us that $(y,x) = \bar{(x,y)}$ for all $x,y \in V$. So, $(x,x) = \bar{(x,x)}$ for all $x \in V$. This shows that $(x,x)$ must be real. So we don't need to modify the third condition. – littleO Dec 21 '20 at 20:59
@Saikat Yes, that is correct. – José Carlos Santos Dec 21 '20 at 21:39

Problem with the Definition of Inner Product Space.

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