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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

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  • $\begingroup$ You haven't given two complex vectors, so you can't get an inner product of them... $\endgroup$ – Thomas Andrews Sep 17 '14 at 16:26
  • $\begingroup$ Depending on your convention, it is $\sum_k \bar{x_k} y_k$. $\endgroup$ – copper.hat Sep 17 '14 at 16:26
  • $\begingroup$ A good way to visualize this, is to keep in my that the inner product is nothing more than the cosine of the angle between two vectors. This allows us to get a useful geometrical intuition and becomes very helpful once we start dealing with inner product space of functions for instance. $\endgroup$ – Drmanifold Sep 17 '14 at 16:35
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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.

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