Can anybody explain about "inner product"?

Question

I am studying inner product space.

One thing thing that I am trying to understand is, "How you define inner product?"

For example For $\mathbb R^3$, if $ x=(x_1, x_2, x_3),\, y=(y_1,y_2,y_3)$, what is $(x\cdot y)$ ? is is just $x_1y_1+x_2y_2+x_3y_3$ ?

Or I have a complex number such that (1+2i, 3+5i) what would be the inner product of it

Answer 1

Let $V$ be a real vector space. Then, an inner product on $V$ is a map $\mathrm{g}:V\times V\to\mathbb{R}$ such that the following holds:

1. $\mathrm{g}$ is bilinear. That is, if $x,y,z\in V$ and $\alpha\in\mathbb{R}$, then $\mathrm{g}(x+y,z)=\mathrm{g}(x,z)+\mathrm{g}(y,z)$ and $\mathrm{g}(\alpha x,y)=\mathrm{g}(x,\alpha y)=\alpha\mathrm{g}(x,y)$.
2. $\mathrm{g}$ is positive-definite. That is, if $v\in V$, then $\mathrm{g}(v,v)\geq 0$, with equality if and only if $v=0$.
3. $\mathrm{g}$ is symmetric. That is, if $v,w\in V$, $\mathrm{g}(v,w)=\mathrm{g}(w,v)$.

One example of an inner product is the dot product on $\mathbb{R}^n$. Another example of an inner product on $\mathbb{R}^n$ might be $$\mathrm{g}:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R},~(v,w)\mapsto v^TAw,$$ where $v^T$ is the transpose of the column vector $v$ and $A$ is a matrix such that $A=A^T$ and all eigenvalues of $A$ are positive.

For complex numbers, we just exchange symmetry for conjugate symmetry.