Find an explicit orthonormal basis of the separable Hilbert space

by applying the Gram-Schmidt procedure to the polynomials [math]f_n=x^n[/math], with [math]n\in\mathbb N[/math].

Develop a theory of projections, isometries and symmetries inside [math]B(H)[/math], notably by examining the validity of the formula

when talking about projections, and also by taking into account the fact that

does not necessarily hold in infinite dimensions, when talking about isometries.

Prove that for the usual matrices [math]A,B\in M_N(\mathbb C)[/math] we have

where [math]\sigma^+[/math] denotes the set of eigenvalues, taken with multiplicities.

Prove that an operator [math]T\in B(H)[/math] satisfies the condition

for any [math]x\in H[/math] precisely when it is positive in our sense, [math]\sigma(T)\subset[0,\infty)[/math].

Clarify, with examples and counterexamples, the relation between the eigenvalues of an operator [math]T\in B(H)[/math], and its spectrum [math]\sigma(T)\subset\mathbb C[/math].

Assuming that an operator [math]T\in B(H)[/math] is normal, [math]TT^*=T^*T[/math], apply the Gelfand theorem to the [math]C^*[/math]-algebra that it generates

in order to deduce a diagonalization theorem for [math]T[/math].

Develop a theory of noncommutative geometry, by formally writing any [math]C^*[/math]-algebra, not necessarily commutative, as

with [math]X[/math] being a “compact quantum space”, and report on what you found.