A Hilbert space is separable if and only if it admits a countable orthonormal basis
50946
Separable Hilbert space have a countable orthonormal basis
51313
Extension of linearly independent set to a basis in an infinite dimensional vector space
95635
$\ell^{\infty}(\mathbb N)$ is not a separable space
92202
Linear separability for a sum of kernel functions
78648
Find an orthonormal basis for the subspace of $\mathbb R^4$
95481
The set of self-adjoint operators over a Hilbert space doesn't form a lattice
133609
orthonormal system in a Hilbert space
30017
Orthogonality and linear independence
16692
a representation condition of Hilbert space
78997
If $L/K$ normal and $H = \operatorname{Aut}(L/K)$, then $L/L^H$ is separable and $L^H/K$ is purely inseparable.
119831
bounded operator $T$ is not compact then there exists an orthonormal sequence $e_n$ and $d>0$ such that $\|T(e_n)\|>d$ for all $n\in\Bbb{N}$?
40820
Weak convergence in separable Hilbert spaces
135650
Sequences in Hilbert spaces
144151
An orthonormal set cannot be a basis in an infinite dimension vector space?
87983
Examples of Separable Spaces that are not Second-Countable
59055
Diagonalizable transformation restricted to an invariant subspace is diagonalizable
116632
Gram-Schmidt in Hilbert space?