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So good morning, welcome to lecture 10. We're working towards the spectral theorem that will be the topic or the proof of the spectral theorem will be the topic of the next lecture.

However, today I will start by stating the full spectral theorem and in there will be some constructions that I will then explain how they're performed.

But today will only come to the proof, which will then actually just be a corollary from our constructions of the inverse spectral theorem best way.

I don't know whether you find in literature the inverse spectral theorem, it's rather trivial, but well that's my pedagogical twist here.

So in this lecture today we're going to develop the technology to state and in the next lecture, so that would be lecture 11,

how to prove the spectral theorem.

The spectral theorem has a very simple form. It says for every self-adjoint operator, operator A defined on a dense domain into the Hilbert space we're looking at,

there is a projection valued measure and what that is we're going to make more explicit in this lecture and some more.

There is a so-called projection valued measure and I'm strongly against acronyms,

but here let's call this a PVM, a projection valued measure, because this is so much writing on the blackboard.

There is a projection valued measure, what is that?

That is a map that takes you from the Borel sigma algebra over the real numbers, so the sigma algebra that is generated by the open sets into the bounded maps on the Hilbert space,

a projection valued measure and there are certain extra conditions which I'm going to state in a second, such that, well, and for every self-adjoint operator A,

so this guy gets the name PA because it's the projection valued measure constructed from A and how we will construct that we will learn,

such that we can write the operator A in the following fashion, which you will not understand right now, but you will at the end of today's lecture.

It can be written as the integral, this still looks good, over the real line of the identity map on R,

where I must say that here by identity on R I mean something slightly more general, I mean the embedding of R into C,

so that takes a real number lambda and embeds it into the complex numbers, well, but still the identity, identity R dPA.

Now, you already know how this Lebesgue integral is defined, if the P was not a projection valued measure,

but if this was, were the real numbers or the complex numbers, then you would know how to define this, the only difference is that it now takes values in the bounded maps,

I'll explain this nevertheless in detail. That's it, that's the spectral theorem, so here we have this,

this is that, and you remember, so this is really just another notation, this means nothing different,

we often don't want to write an abstract name for a map here, we rather give the prescription, the function values of the map,

and in that spirit I just give here id R of lambda is just lambda, and then you know that we agreed already for measures to also,

to also write this in this fashion, alright, you remember that, so please compare, compare to the lecture on measure theory,

on measure theory, this is just notation, this is more often met in the literature than this, however I find this a little clearer,

so this is the spectral theorem, and this is invaluable in quantum mechanics, well in fact you can't really do it without the spectra, oh, okay, wow,

okay, the spectral theorem, okay, so now you see there is, that sounds terribly non-constructive, it sounds like an existence result,

and as it is with existence results, they can be very useful in theory, but they're very little useful in practice,

because unless you can construct this P A given an self-adjoint operator, you know you can write like this,

but you would really like to know how to construct these P As, now the thing is, it is constructive,

the good news is, but there are only good news, okay, there's only good news, so the theorem is in fact constructive,

for we will see that from given self-adjoint, that's important, self-adjoint A,

we can construct P A along the following three steps,

and I'll give them right now, because then you know what the whole thing is about,

and then we will start the other way around and start developing the definitions and proofs,

such that next lecture we'll be ready to prove that it works like this, okay,

but first I give you the application how it works and how you will use it in practice, okay,

so the first step, step one, you construct the real valued, so before we construct a projection valued measure,

you construct the real valued Borel measure, what is that, well construct a real valued Borel measure for each psi in H,

so we will have one projection valued measure later on, a unique, aha, I should even emphasize, there is a,

in the spectral theorem, a unique, a unique projection valued measure, that's important,

they will be associated with every A, a unique projection valued measure,

well the projection valued measures in order to work with them, we break them down to real valued measures first,

but then we have to specify a psi, so the real valued Borel measure will have the name mu psi,

and it will depend on the choice of psi, but we'll construct it for any psi, that's the point, for each psi,

what is such a measure, well it's a measure, it has the name mu psi, it's constructed from a psi,

it takes a Borel set in the real number, so a measurable subset of the real numbers,

and it will produce a real number, that's why it's real valued, whereas this guy was operated, bounded operator valued,