A Hilbert Space is generalisation of euclidean space.

A Hilbert Space H is an Inner Product Space that has an induced Distance Function (Norm) like this:

|x|:=√{(x,x)} (with x∈H)

Since H is Inner Product Space, it holds that:

(x,y)^*=(y,x) swapping of inner product operands is complex conjugation of the result
(x,a⋅y+b⋅z)=a⋅(x,y)+b⋅(x,z) Linearity in the second argument
(x,x)≥0 positive definiteness (only when both arguments are the same)
(a⋅y+b⋅z,x)=a^*⋅(y,x)+b^*⋅(z,x) Antilinearity in the first argument
|(x,y)|≤|x|⋅|y| Cauchy-Schwarz inequality
∀u:∑\limits_{k} |u_k|<∞ Absolute Convergence of all vectors

Dirac Hilbert Space

In Quantum Mechanics, a specific Hilbert Space of Square Integrable Functions L² is used:

(a,b):=∫\limits_{-∞}^{∞} a^*[x]⋅b[x]⋅dx

Note that the above, in general, is a complex number

(a,a)<∞ This is a real number.

A vector in this space is written as:

|ψ⟩ Ket Vector

A Hilbert Space is a linear vector space, so:

|φ⟩+|ψ⟩=|φ+ψ⟩
|φ⟩+|ψ⟩=|ψ⟩+|φ⟩ Commutativity
|φ⟩+|ψ+ξ⟩=|φ+ψ⟩+|ξ⟩ Associativity
c⋅|φ⟩=|c⋅φ⟩ Scalar multiplication
c⋅|φ+ψ⟩=c⋅|φ⟩+c⋅|ψ⟩ Distributivity

The Hilbert Space is complete, so every linear combination of vectors in a Hilbert space is again a vector in the same Hilbert space.

The Bra Vector is the complex conjugate of the Ket Vector, an element of the dual space of H, also an element of the same Hilbert space H:

|ψ⟩^†=⟨ψ|
(C|ψ⟩)^†=⟨ψ|C Reversal of Operators

Reason for the reversal of operators:

(a,L b)=(L^† a,b) Definition of Adjoint Operator L^† to L

Components

Let |ψ_n⟩∀n be a (for example orthonormal) basis. Then because of the completeness, any vector |φ⟩ can be written as a linear combination with respect to that basis:

|φ⟩=∑\limits_n |ψ_n⟩⋅c_n

Then the coefficients can be determined by completing the scalar product over the equation, just as one would within the Euclidean Space

⟨ψ_m|φ⟩=⟨ψ_m|∑\limits_n |ψ_n⟩⋅c_n
⟨ψ_m|φ⟩=∑\limits_n ⟨ψ_m|ψ_n⟩⋅c_n

If the basis is orthonormal, it follows that

⟨ψ_m|φ⟩=∑\limits_n δ_{mn}⋅c_n
⟨ψ_m|φ⟩=c_m
⟨ψ_n|φ⟩=c_n
|φ⟩=∑\limits_n |ψ_n⟩⋅⟨ψ_n|φ⟩

We say the projection operator P is (without sum):

P_n\:x:=|ψ_n⟩⋅⟨ψ_n|\:x

Projection is idempotent, projecting multiple times in a row does not change the result.

On the other hand, we say that the identity operator is:

\mbox{id}\:x=∑\limits_n |ψ_n⟩⋅⟨ψ_n|\:x

By applying this operator both to the left and to the right side of another operator L, we get its components with respect to a basis:

\mbox{id}\:L\:\mbox{id}=\mbox{id}\:L\:∑\limits_n |ψ_n⟩⋅⟨ψ_n|
\mbox{id}\:L\:\mbox{id}=∑\limits_m |ψ_m⟩⋅⟨ψ_m|\:L\:∑\limits_n |ψ_n⟩⋅⟨ψ_n|
\mbox{id}\:L\:\mbox{id}=∑\limits_m |ψ_m⟩⋅∑\limits_n ⟨ψ_m|L|ψ_n⟩⋅⟨ψ_n|
\mbox{id}\:L\:\mbox{id}=∑\limits_m ∑\limits_n ⟨ψ_m|L|ψ_n⟩⋅|ψ_m⟩⋅⟨ψ_n|
L_{mn}:=⟨ψ_m|L|ψ_n⟩
\mbox{id}\:L\:\mbox{id}=∑\limits_m ∑\limits_n L_{mn}⋅|ψ_m⟩⋅⟨ψ_n|

L_{mn} are the components with respect to the basis ψ_....