Some questions on coherent states and corresponding Hilbert spaces. Reproducing kernal

I have a few questions related to coherent states. I use this source https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf.

Using standart inner product $$\langle\cdot|\cdot\rangle$$ in Hilbert space $$L^2(\mathbb{R})$$ we can find the value of coherent states and Fock states at a point, if I understand correctly, as shown in $$(5.126)-(5.128)$$ by the link above. But directly calculating, I get a different result $$\langle x|m\rangle=\exp^{-|x|^2/2}\sum^\infty_{n=0}\frac{x^n}{\sqrt{n!}}\,\langle n|m\rangle=\exp^{-|x|^2/2}\frac{x^m}{\sqrt{m!}}.$$ Where am I mistaken? This is the first question.

It is also interesting to me, Is the set of coherent states dense in $$L^2(\mathbb{R})$$? Or maybe it forms some Hilbert subspace? Is it possible to build a reproducing kernal in it?

• You should consult the canonical textbook by Perelomov on coherent states for your questions. – ZeroTheHero Feb 4 at 14:40
• Note that (126) and (127) are NOT coherent states, but the standard h.o. wavefunctions in the position representation. – ZeroTheHero Feb 4 at 14:43
• This book? "Generalized coherent states and some of their applications A M Perelomov" – Иван Петров Feb 4 at 15:54
• That is, it is not a inner product between the coherent state and wavefunction in 126-127? – Иван Петров Feb 4 at 16:04