Merge S106 into S30, resolve #829

[yhx-12243]

See discussion in #829.

[prabau]

I am not saying this is the wrong approach, but you should get agreement in issue #829 first.
At this point, nothing has been decided yet.
I.e., need @StevenClontz input.

[StevenClontz]

Let's table this PR for now and discuss in #829.

[prabau]

Preview now available on branch yhx-12243/pr863-preview.

[prabau]

Overall the intended meaning is there. I think we can take the opportunity to clean up a few things in the README file that are not completely clear.

Would also appreciate input from @GeoffreySangston, @StevenClontz and others on the suggestions below.

Primary name: There are many non-homeomorphic Hilbert spaces. What we want is the separable ones here. How about making Hilbert space $\ell^2$ the primary name?

Aliases: Suggestions:

• remove $\ell^2$ (already covered as part of the primary name)
• combine $\mathbb R^\omega$ and Frechet space into Fréchet space $\mathbb R^\omega$
• remove $C(I)$ as unclear out of context
• keep $C[0,1]$, but only because it's the name from Steen & Seebach. Wouldn't $C([0,1])$ be more usual?

For the text itself, we have a progression with first paragraph = separable Hilbert spaces, second = Anderson thm, third = Anderson-Kadec. So we can rephrase things a little to make that clearer, and also organize things slightly better. How about something like:

More generally, any infinite-dimensional separable Banach space or separable Fréchet space is homeomorphic to $\mathbb R^\omega$. See {{wikipedia:Anderson–Kadec theorem}}.

In particular, the following are all homeomorphic as topological spaces:

• $\mathbb R^\omega$
• Sequence spaces $\ell^p$ ($1 \leq p < + \infty$)
• Spaces $L^p([0,1])$ ($1 \leq p < + \infty$) of Lebesgue measurable functions with finite $L^p$-norm
• the Banach space $C([0,1])$ of real-valued continuous functions on $[0,1]$ with the supremum norm

(maybe you could add wikipedia links for each of the above for convenience?)

And change the final paragraph to:

These cover counterexamples ... ... ... from {{...}}.

(don't need to repeat the link for S&S twice)

[prabau]

@StevenClontz I did not know this, but apparently a link like {{wikipedia:Anderson–Kadec theorem}} works even with blanks in the middle and displays the right thing without being in the refs: section. See the preview on branch yhx-12243/pr863-preview.

[StevenClontz]

Would also appreciate input from @GeoffreySangston, @StevenClontz and others on the suggestions below.

I agree (and suggested a few specific changes along some of those lines).

[StevenClontz]

@StevenClontz I did not know this, but apparently a link like {{wikipedia:Anderson–Kadec theorem}} works even with blanks in the middle and displays the right thing without being in the refs: section. See the preview on branch yhx-12243/pr863-preview.

Yep. What happens:

• First, you can always insert a {{tag:id}} even if there isn't a relevant refs: for it. (In fact, it's related to why pi-Base doesn't display the name of the work in the description when it's given in the refs.)
• Wikipedia's server redirects the URL https://en.wikipedia.org/wiki/Anderson%E2%80%93Kadec%20theorem and converts it to https://en.wikipedia.org/wiki/Anderson%E2%80%93Kadec_theorem (where %E2%80%93 is the – literal and %20 is the literal).

But we should use the canonical wikipedia ID rather than rely on redirection, so we should use an underscore instead of a space.

[GeoffreySangston]

Would also appreciate input from @GeoffreySangston

I'm sorry that I never saw this until now. I'm not sure I have anything valuable to add to this particular suggestion other than it looks nice to me.