Space of square-summable sequences
From Maths
Definition
The space of square-summable sequences, denoted [ilmath]l_2[/ilmath], is the space of all (countable) sequences of either complex, or real numbers[1]. That is:
- [math](x_n)_{n=1}^\infty\subset\mathbb{R}[/math] or
- [math](x_n)_{n=1}^\infty\subset\mathbb{C}[/math]
With the property of:
- [math]\sum_{n=1}^\infty\vert x_i\vert^2< \infty[/math]
Usual inner product
This space is usually equipped[1] with the following inner product:
- For [ilmath]x,y\in l_2[/ilmath] we define [ilmath]\langle x,y\rangle:=\sum^\infty_{n=1}x_i\overline{y_i}[/ilmath]
Proving this requires things like Holder's inequality (with the funny o) and is something I need to do:
TODO: Page 9 is a start of the first ref
