## Inner Product Space PDF.

### July 19, 2010

So, I wanted to proceed onwards towards some pretty cool mathematics (and, finally, get through basic linear algebra) by introducing the Gram-Schmidt ortho-normalization process and some really sweet consequences (that actually really surprised me!), but it occurred to me that I’d need to introduce norms and inner products as well as prove a butt-load of things about them.

Because I am terribly lazy, I am not going to do this.  Instead, I read through a number of inner product introductions (which are all basically the same) to find one that was well-written.  The one that I’ve picked to show ya’ll is from G. Keady, from the University of Birmingham.  It is in pdf form, and it is available here (warning, pdf!).

With the possible exception of ultra-brevity (orthonormal is abbreviated ON, and that’s kind of weird to get used to) and some of the things at the end of the paper, this is a 3-page introduction and, partially because of this, is very readable.  You should not need any math besides what we’ve already covered in this blog.

We’ll give examples of normed vector spaces and inner product spaces later, but we’ll definitely be using the inner product space of continuous (real) polynomials on the interval $[0,1]$, which has the inner product

$\displaystyle\langle f, g\rangle = \int_{0}^{1} f(x)g(x)dx$

This will come in handy later, so remember it!