In this PDF https://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf on page 14, the author says right after equation (1.11)
Note that by using a basis of Hilbert-Schmidt orthogonal unitaries $\{U_j\}_{j=1,...,d^2}$ we can construct an orthonormal basis of maximally entangled states
In this document, he considers the Hilbert space $\mathcal{M}_d(\mathbb{C})$ where each element is a $d \times d$ matrix. The scalar product between two elements of this Hilbert space is:
$$ \langle A | B \rangle = Tr(A^{\dagger} B)$$
My question is: How do we know we can have a basis of $\mathcal{M}_d(\mathbb{C})$ composed of unitary matrices ? For me asking the element of the basis are unitary matrices is a very strong condition. How is it possible ?
I would like answer not too technical, my main focus is quantum information, not general theory around Hilbert spaces. I have good basis in linear algebra but for example the Hilbert Schmidt inner product for matrices is a new concept for me.