Quantum mechanics and rigorous math I was reviewing a little of quantum mechanics in a rigorous way, so i realized there is a lot of concepts similar in words but different in its meanings, i would appreciate any help to understand it:
Kets -> It is the "physics entity" you are measuring: Momentum, energy, etc...
Operators -> What you actually use to get the results of Momentum, energy, etc...
Eigenkets -> The basis, the possible states the operator can have: exp: Spin up or spin down
Eigenvalues -> The possible values you can get.
Eigenfunctions -> Reserved to wave functions
Eigenvectors -> ?
State vector -> ?
 A: Very short and brief summary of quantum mechanics, please comment if there are mistakes
Ket vs. State
Each equivalence class of kets (vectors) which differ from each other by a (multiplicative) complex number is called a ray in Hilbert space (which is a complete inner product space). Each ray represents a state uniquely.

The state has many representations, the most famous one is the wave function, the so-called position space representation. Another one is the momentum wave function, in momentum space. It is possible to switch between them.

Hermitian operators (linear maps on the Hilbert space) represent observables, the eigenvalues of such an operator are the possible outcomes of a measurement of the corresponding observable, the quantum system takes on a corresponding eigenstate (eigenket) immediately after measurement.

A complete set of compatible observables (C.S.C.O) is a set of observables $A_1,A_2,...,A_n$ whose eigekets can constitute an orthonormal basis of the Hilbert space, such that every eigenket (basis vector) corresponds to a unique tuple $(a_{i_1}^{(1)},a_{i_n}^{(2)},...,a_{i_n}^{(n)})$ of eigenvalues of the C.S.C.O. Each basis vector is denoted $|a_{i_1}^{(1)},a_{i_n}^{(2)},...,a_{i_n}^{(n)}\rangle$. The general state is an "infinite (complex) linear combination" of this basis, The evolution of the state is determined by Schrödinger equation.

The probability of getting some eigenvalue (as an outcome of a measurement of some observable) is (in case of degeneracy) the (sum of) the squares of the magnitude of the coefficients (of the previous linear combination) that correspond to eigenkets of the considered eigenvalue.
A: Kets
In quantum mechanics the possible states of the system are elements of a separable, projective Hilbert space (i.e. two states differing by an overall complex constant are equivalent). The kets e.g. $|\psi\rangle$ are elements of the Hilbert space $\mathcal H$, the bras e.g. $\langle \psi|$ are elements of its dual, $\mathcal H^*$.

Operators
Each physically measurable quantity is associated with a Hermitian operator, e.g. position $\hat x$, momentum $\hat p$, energy (aka. the Hamiltonian) $\hat H$ etc.

Eigenkets
These are eigenvectors of the operators that correspond to the observables. Note that since we are dealing with Hermitian operators the eigenkets corresponding to distinct eigenvalues are orthogonal.
*Note, terms often used synonymously with "eigenket" in physics: Eigenstate, eigenvector, eigenfunction.

Eigenvalues
The spectrum of the operators that correspond to observables are the (only) possible values of that quantity that can be obtained upon measurement of the observable in question. Given a system in a state $|\psi\rangle$, the probability of obtaining the value $a$ and (necessarily) finding the system in the associated eigenstate $|a\rangle$ after measuring the observable $\hat A$ is given by: $$\mathrm{Probability}=|\langle a|\psi\rangle|^2 \tag{1}$$

Eigenfunctions
There is an isomorphism $\mathcal H\cong L^2(\Bbb R)$ that allows us to associate elements of our Hilbert space with elements of the space of square integrable functions, elements of the latter are called wavefunctions.

Additional note: You can be more rigorous than this, rigorous quantum mechanics is possible to do but requires a significant amount of heavy lifting and a fair amount of machinery that is not introduced even in fairly advanced courses.
