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You're most welcome. The rest of the expansions seem right to me. As to proof 4.3, I see what you're doing there, and it's correct too, AFAICS. The only glitch is for these kind of proofs is that it's perhaps best to drop Dirac's notation, because it kind of stands in the way of distinguishing the vector, the operator, and the action of the inner product more clearly, so you wouldn't have to use the --somewhat awkward, IMO-- double parenthesis on your last line. So, for example, I would write something like, for every \( w \), \( v \) in \( \mathscr{H} \) (the Hilbert space of states), \[ \left( w, P\right) = \left( P^{\dagger}w,v \right) \] (That is just a definition of \( P^{\dagger} \), of course) I would also write, \[ v=v_{perp}+v_{parallel} \] etc, with, \[ Pv_{perp}=0 \] \[ Pv_{parallel}=v_{parallel} \] for an arbitrary vector \( v \). And then, as you say, \[ \left( w, Pv \right) = \left( w,w_{\parallel} \right) = \left( w_{\parallel}, v_{\parallel} \right) = \left( w_{\parallel}, v \right) = \left( Pw, v \right) \] So indeed \( P^{\dagger}=P. To me, it's a bit more transparent with this notation, but I understood what you meant, and if you think about it we're saying the same thing. The devil is in the details, as they say. In infinite dimension one would have to be much more careful than this, but I don't have the chops for it. 😊 Ok. Something got messed up in the LaTeX rendering, I'm afraid... I hope you can see what it is. I always have to be very careful that my text is not interpreted as rich text at some point (eg, when the editor refreshes) so the rendering is messed up. It might have to do with the software at my end. I dunno.

Let's consider a direct sum decomposition of the space V = S ⊕ S⊥, where S and S⊥ are orthogonal, and let's consider a projection operator P of V onto S. Take two vectors (|v〉+|v⊥〉) and (|w〉+|w⊥〉) in V, where |v〉 and |w〉 are in S and |v⊥〉 and |w⊥〉 are in S⊥. By definition, P(|v〉+|v⊥〉)=|v〉. (〈w|+〈w⊥|)(P(|v〉+|v⊥〉)) = (〈w|+〈w⊥|)(|v〉) = 〈w|v〉. OTOH, by definition, P(|w〉+|w⊥〉)=|w〉. ((〈w|+〈w⊥|)P)(|v〉+|v⊥〉) = (〈w|)(|v〉+|v⊥〉) = 〈w|v〉. Thus, (〈w|+〈w⊥|)(P(|v〉+|v⊥〉)) = ((〈w|+〈w⊥|)P)(|v〉+|v⊥〉), i.e., P is its own adjoint.

I thought the real savings are in all the systems needed to keep humans alive.

Screwing is sometimes more literal than I had realized.

If Gazans were spread out, they have about 150m2 of land each. It has one of the highest population densities in the world. I did a back of the envelope sum yesterday. Not all of that 140km2 will be habitable to hold the 2 million population. They have even less space now the high-rises are getting blown up. Compressing people like that can only perpetuate animosity. Likud want to wipe out/evict Palestinians, so that they can realize their biblical colonial fantasy. This, I think, is their ultimate raison de etre and where their current actions are leading. Philosophically, where one has peace of mind, I equate that with a proper sense of freedom.

You went into Matrix-mode and used your unknown yet Neo-like features..

Male ducks have clockwise corkscrew-like penises. When a female duck is not interested in the male's advances she tightens up her anti-clockwise corkscrew-like vagina to prevent the male from achieving his full (amazingly long) erection. Mother nature certainly has a sense of humor. https://www.sciencefocus.com/nature/duck-penis-corkscrew

The kinetic energy of something makes it come out. If you look at the math one can note by the change in energy is done by many variances. Every change causes the change in energy. All the deltas are one. Everybody can do this math. So you have to put it together.

It is absurd to think that this is the FIRST time ever to have existed. But as the discussion shows, even if we discover anything it is in the past. Nothing can be done about it. However, we live in a time system.

It looks like there are circular grooves. Maybe we give it a new word instead of grooves.