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Please note I have two questions regarding Rudin RCA 4.17. I have consulted the following related questions and these are follow up questions based on the comments from these questions:

  1. Questions in the Proof of Riesz-Fischer Theorem and Bessel's Inequality (Rudin's RCA)

  2. Is the norm on a Hilbert space always finite?

a) In RCA Definition 4.13: The definition of $\hat{x}$ is given as a complex function defined on the index set $A$ for every $x\in H$ such that $\hat{x}\left(\alpha\right)=\left(x,u\alpha\right)$. How are we to interpret this definition of $\hat{x}$ as used in Theorem 4.17. Specifically referring to the third part of the question from the linked question 1, (How does Theorem 4.14(a) show that $f$ is an isometry of $P$ onto the dense subspace of $\ell^2(A)$ consisting of those functions whose support is a finite subset of $A$?): We have to show that $\left\Vert y_{1}-y_{2}\right\Vert =\left\Vert \hat{y}_{1}-\hat{y}_{2}\right\Vert $ where $y_{1},y_{2}\in H$. We can take $y_{1},y_{2}\in P$ or as linear combinations of the orthonormal vectors. But what about $\hat{y_{1}},\hat{y}_{2}$? How can we represent them?

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b) Can you please clarify why $\left\Vert x\right\Vert ^{2}<\infty$? I am unsure how the definitions given in Rudin RCA regarding a Norm restrict it to be a finite value. The following picture shows the definition of Norm from 4.1, which I think is relevant. Am happy to add the definition of Lp-Norm from chapter 3 if that is needed as well. enter image description here

Please let me know if you need any additional clarifications.

Regarding the comments about complex numbers and infinity ($\infty$) which mention that $\infty$ is not a complex number. Thanks for pointing this out. While this seems like a reasonable assumption to make on which the definition of norm can be based. I am not entirely sure this follows from what is given in Rudin RCA.

We can infer this based on the definition of a simple function given in Chapter 1 (1.16): A simple function is a complex function from whose range we explicitly exclude infinity ($\infty$). This means that a not simple complex function, which maps to the complex plane, can include infinity. Perhaps there is something I am overlooking. Can you please point out or clarify?

enter image description here

texmex
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    $\infty$ is not a real number or a complex number. So $(x,x)$ cannot be $\infty$. – geetha290krm Aug 22 '23 at 05:20
  • Thanks, So we exclude the extended reals from the definition of a Norm. – texmex Aug 22 '23 at 05:22
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    Rudin clearly says that $(x,y)$ is a complex number. – geetha290krm Aug 22 '23 at 05:23
  • @geetha290krm Thanks for pointing this out. While this seems like a reasonable assumption to make on which the definition of norm can be based. I am not entirely sure this follows from what is given in Rudin RCA. Please see the portion I added to the question. From the definition of a simple function, which excludes $\infty$ we can infer that a not simple complex function can include $\infty$. Perhaps there is something I am overlooking. Can you please point out or clarify? – texmex Aug 24 '23 at 02:34

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