- #1

Cauchy1789

- 46

- 0

## Homework Statement

I have posted simular questions a couple of times but now I feel I have a better understanding(hopefully).

Given a Vectorspace M which is defined as a sequence of realnumber [tex]\{r_n\}[/tex] and where [tex]\sum_{r=1}^{\infty} r_n < \infty[/tex]

Show that M has an innerproduct given by

[tex]\langle \{r_n\}, \{p_n\}\rangle = \sum_{n=1}^{\infty} r_n \cdot p_n[/tex]

There is also a question regarding that one needs to show that M is complete with respect to the induced norm [tex]\|\{r_n\}\|_2 = \sqrt{(\sum r_n^2)}[/tex]

## The Attempt at a Solution

First I will try to verify that M is indeed a Vector Space. From the definition Our space M is defined as a sequences of real numbers [tex]\{r_n\}[/tex]. Thus that leads me to think of it as a Euclidian n-space definition. Which if I remember my linear algebra correct leads to the conclusion that our space M satisfies the axioms of a the Vector Space and hence M is indeed a Vector Space.

How is that?

If this is true (if I understand this correctly) then this will lead me to that if [tex]r_n[/tex] is a vector in both M and [tex]\mathbb{R}^n[/tex], and I can choose a scalar called let's call it and choose S, and then I choose a second and third vector in [tex]\mathbb{R}^n[/tex]

and then prove that the axioms of the inner product can be applied to M successfully.

How is that?

Have I understood this correctly?

Cheers

Cauchy