basic things that are good to have in one place

pre-Hilbertian space $\mathcal{H}$ is a $ℝ$ or $ℂ$ vector space with $⟨⋅|⋅⟩ : \mathcal{H} ⨯ \mathcal{H} → ℂ$ such that $∀x,y,z ∈ \mathcal{H}, ∀a,b ∈ ℂ$
1. $⟨x|y⟩ = \overline{⟨y,x⟩}$
2. $⟨ax+by|z⟩ = a⟨x|z⟩ + b⟨y|z⟩$
3. $⟨x|x⟩ = 0 \iff x=0$ Hilbert space $\mathcal{H}$ is
1. a pre-Hilbertian space
2. a Banach space (ie complete space) for the norm $||x|| = \sqrt{⟨x|x⟩}$ The very useful dirac notation allows us to denote $x∈\mathcal{H}$ as $|x⟩$ to be seen as the "half right" of the scalar product, as if $x$ exists solely to at one point get projected on some other vector. Then $⟨x|$ is the "conjugate transpose" in finite dimension, it can be seen as the "other half" scalar product.
The conjugate part appears because of point 1. in pre-Hilbertian space definition. $\{e_1,e_2,...\}$ such that $⟨e_i|e_j⟩=δ_{i,j}$ is a hilbertian basis if $∀ψ∈\mathcal{H},|ψ⟩ = ∑_{n \geq 1}|e_n⟩⟨e_n|ψ⟩$ If $\mathcal{H}$ has a hilbertian basis (a countable set as basis), then it is said to be separable (which is very similar to what we have in topology) If $\mathcal{H}$ has a hilbertian basis, then it is isomorphic to $l_2:=\left\{ ψ = (ψ_1,ψ_2,...) \middle| ∑|ψ_n|² < ∞ \right\}, ⟨φ|ψ⟩_{l_2}=∑_n\overline{φ_n}ψ_n$ just consider $σ:\mathcal{H}→l_2$ defined as $σ(ψ) = (⟨e_1|ψ⟩,⟨e_2|ψ⟩,...)∈l_2$ $ρ:\mathcal{H}_1→\mathcal{H}_2$ is a bounded operator if $∃C>0$ such that $∀ψ∈\mathcal{H}_1, ||ρψ||_2 \leq C ||ψ||_1$
then the norm is $||ρ|| = \sup_{0≠ψ∈\mathcal{H}_1} \frac{||ρψ||_2}{||ψ||_1}$
the set of bounded operators is denoted $\mathcal{B}(\mathcal{H})$ the adjoint of $ρ:\mathcal{H}→\mathcal{H}$ bounded is $ρ^∗:\mathcal{H}→\mathcal{H}$ the unique operator such that $∀φ,ψ∈\mathcal{H}, ⟨φ|ρψ⟩=⟨ρ^∗φ|ψ⟩$ $ρ$ is auto-adjoint if $ρ=ρ^∗$ Riesz theorem : let $l:\mathcal{H}→ℂ$ a linear bounded form, then $∃!ψ_l∈\mathcal{H}$ such that $∀φ∈\mathcal{H}, l(φ) = ⟨ψ_l|φ⟩$ The consequence is with $ρ:\mathcal{H}→\mathcal{H}$ bounded that $l_φ(ψ):=⟨φ|ρψ⟩$ being a linear bounded form, $∃!η_φ∈\mathcal{H}$ such that $⟨φ|ρψ⟩ = ⟨η_φ|ψ⟩$, which will make us want to have $ρ^∗φ=η_φ$ let $l:D⊂\mathcal{H}→ℂ$ be a linear bounded form on $D$ that is dense in $\mathcal{H}$, then $l$ has a unique extension over $\mathcal{H}$ $ρ:\mathcal{H}→\mathcal{H}$ bounded, $\mathcal{H}$ separable.
⋅ $ρ$ is positive if $∀x∈\mathcal{H}, ⟨x,ρx⟩\geq0$
⋅ $ρ$ is trace class if $Tr(ρ):=∑_n⟨e_n|ρe_n⟩ < ∞$
⋅ $ρ$ is projection if $ρ²=ρ$
⋅ $ρ$ is orthogonal projection if $ρ²=ρ$ and $ρ^∗=ρ$
⋅ $ρ$ is unitary if $ρ^∗ρ = 1 = ρρ^∗$ If $A$ is trace class, and $B$ just bounded, then both $AB$ and $BA$ are trace class with $Tr(AB)=Tr(BA)$ $A:D(A)⊂\mathcal{H}→\mathcal{H}$ is densely defined if $D(A)$ is dense in $\mathcal{H}$ $A:D(A)⊂\mathcal{H}→\mathcal{H}$ is symmetric if $∀ψ,ϕ∈D(A), ⟨ψ|Aϕ⟩ = ⟨Aψ|ϕ⟩$ $A$ symmetric implies that $Aψ=λψ$ must have $λ∈ℝ$ let $A:D(A)⊂\mathcal{H}→\mathcal{H}$ a linear operator be densly defined, then the adjoint $A^∗: D(A^∗)⊂\mathcal{H} → \mathcal{H}$ is defined with $$ \ca{ D(A^∗):=\{ψ∈\mathcal{H} | ∃!η_ψ∈\mathcal{H} \text{ such that } ∀α∈D(A), ⟨ψ|Aα⟩=⟨η_ψ|α⟩\} \\ A^∗ψ:=η_ψ } $$ this definition is a bit painful, but it just states that the adjoint is what we usually expect, but we want to make sure that it is properly defined (ie $η_ψ$ is unique) : infinite dimension space can cause surprises. TODO : insert a messed up example $A:D(A)→\mathcal{H}$ linear densly defined is auto-adjoint if
1. $A$ is symmetric
2. $D(A)=D(A^∗)$ let $A:D(A)→\mathcal{H}$ and $B:D(B)→\mathcal{H}$ be linear operators. $B$ is extension of $A$ if
1. $D(A)⊂D(B)$
2. $Bψ=Aψ, ∀ψ∈D(A)$ $A$ symmetric densly defined operator. Then $A^∗$ is an extension of $A$ which means
1. $D(A)⊂D(A^∗)$ (so $D(A^∗)$ is dense too !)
2. $A^∗ψ = Aψ, ∀ψ∈D(A)$ $A$ symmetric and densely defined is essentially auto-adjoint if $(A^∗)^∗ \equiv A^{∗∗}$ is auto-adjoint let $A$ be essentially auto-adjoint, then $∃!$ auto-adjoint extension which is $A^{∗∗}$ if $A$ symmetric densly defined, then $A$ is essentially auto-adjoint $\iff Im(A+i)$ and $Im(A-i)$ are dense in $\mathcal{H}$ $A:D(A)→\mathcal{H}$ linear operator. the résolvante set of $A$ is $$ρ(A):=\left\{λ∈ℂ \middle| ∃B:\mathcal{H} → \mathcal{H} \text{ bounded such that } \al{ 1. && Im(B) ⊂ D(A) && ∀ψ∈\mathcal{H} && (A-λ)Bψ=ψ\\ 2. && && ∀ψ∈D(A) && B(A-λ)ψ=ψ\\ } \right\}$$ This definition is a barbaric way to state that $(A-λ)$ is invertible with inverse $B$ The spectrum of $A$ is $σ(A) := ℂ \backslash ρ(A)$ if $λ$ eigenvalue of $A$ then $λ∈σ(A)$ $ρ(A)$ is open, $σ(A)$ is closed
if $A$ auto-adjoint then $σ(A)⊂ℝ$

the real deal

state $ρ:\mathcal{H}→\mathcal{H}$ is a linear operator that is positive, trace class with $Tr(ρ)=1$ if $ρ$ positive then $ρ$ is auto-adjoint. So a state is auto-adjoint Hilbert-Schmidt theorem (spectral theorem) let $\mathcal{H}$ be separable and $ρ:\mathcal{H}→\mathcal{H}$ be a state. Then $∃ \{e_1,e_2,...\}$ a hilbertian basis and $∃λ_n∈ℝ_+$ with $\lim_{n→∞}λ_n=0$ such that $ρe_n=λ_ne_n$ observable $A: D(A)⊂\mathcal{H}→\mathcal{H}$ are self-adjoint linear operator. measure of observable $A$ on state $ρ$ gives the probability $μ_ρ^A(E) := Tr(P_A(E)ρ)$ that the outcome of the experience resides inside $E$ a measurable set (Borel set to be more precise), where $P_A: \mathcal{A} → \mathcal{B}(\mathcal{H})$ maps a Borel set to a PVM measure associated to the observable $A$.