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**1. The problem statement, all variables and given/known data**

Given the vectorspace consisting of a realvalued sequences [tex]\{x_j\}[/tex] where [tex]\sum_{j=1}^{\infty} x_j^2 < \infty [/tex]. Show that M the vectorspace has an innerproduct given by

[tex]\langle \{x_j\}, \{y_j\}\rangle = \sum_{j=1}^\infty x_j \cdot y_j [/tex]

**2. Relevant equations**

Since [tex]\{x_j\}[/tex] defines every possible vector component in M, then isn't that equal to that the square sum of every possible realvalued vectorcomponent of the M can be written as [tex]\sum_{j=1}^{\infty} x_j^2 \leq \infty (\mathrm{max} |x_j|)^2 = \infty, j = 1, \ldots, \infty[/tex].?

**3. The attempt at a solution**

All possible vector of either [tex]\{x_j\}[/tex]or [tex]\{y_j\}[/tex] are considered to be real valued, thus the definition of the inner product from Linear Algebra is true, hence by condition (1) of the innerproduct

[tex]\langle \{x_j\} \cdot \{x_j\} \rangle = x_j ^2 [/tex] and thus the innerproduct of every component in either x or y can be written as

[tex]\langle {x_j \} \cdot \{y_j \} \rangle = x_j \cdot y_j, j = 1, \ldots, \infty[/tex]

Have I covered all what is required of me in showing the above?

Sincerely

Cauchy

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