@wini_boson , sorry to interrupt you but how come you mentioned me ..lol its been 4 months since i left the OS...but as far as your question is concerned there are three basic properties of hermitian operators and that will answer your first question
--> Hermitian operators can be flipped over to the other side in inner products
-->Hermitian operators have only real eigenvalues
--> Hermitian operators have a complete set of orthonormal eigenfunctions (or eigenvectors)
now second part of your question is quite lengthy to solve here but still i hope it helps you out, most of it is maths which you can ask @sarahusher , if you have the confusion and i hope you can interpret this maths in terms of physics :D
Maths-->
to form a basis of a vector space you need to define a subset of the elements of vectors that are linearly independent and vector space span V...
\[V= V _{1} i+V _{2} j+V _{3} k.....\] V1, V2 , V3 are the elements
and therefore, your vector forms the basis iff V can be expressed as
\[V=a _{1}V _{1}+a _{2}V _{2}+.....\], where a1, a2... are elements of base field
A vector space V will have many different bases, but there are always the same number of basis vectors in each of them. The number of basis vectors in V is called the dimension of V. Every spanning list in a vector space can be reduced to a basis of the vector space.
When a vector space is infinite dimensional, then a basis exists, as long as one assumes the axiom of choice. A subset of the basis which is linearly independent and whose span is dense is called a complete set, and is similar to a basis. When V is a Hilbert space, a complete set is called a Hilbert basis.
*footnote--> sorry couldnt think of an example at 5.20 in the morning :P ...hope that helps you out :D i shall try to get an example for you :D