Space of square-summable sequences

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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



References

  1. 1.0 1.1 Functional Analysis - George Bachman and Lawrence Narici