Let $||\cdot||_F$ and $||\cdot||_2$ be the Frobenius norm and the spectral norm.

I'm reading Ji-Guang Sun's paper 'Perturbation Bounds for the Cholesky and QR Factorizations' from BIT 31 in 1991. While deriving the perturbation bound on Cholesky factorization $A=LL^T$ with $A\in\mathbb{R}^{n\times k}$ of rank $k$, he begins with the perturbed matrix $A+E$ and its Cholesky factorization.

$A+E=(L+G)(L+G)^T$

Expanding the RHS and subtracting $A$ from both sides, we have $E = GL^T + LG^T + GG^T$. Then he claimed in (2.12) of his paper that

$||E||_F\leq 2||L||_2||G||_F + ||G||_F^2$.

If I read $||L||_F$ instead of $||L||_2$, everything seems perfectly normal to me by the following three properties of any matrix norm.

- $||X^T||=||X||$ for any matrix norm
- subadditivity: $||X+Y||_p\leq||X||_p + ||Y||_p$
- submultiplicity: $||XY||_p\leq ||X||_p\cdot||Y||_p$ for $p=2$ or $F$

But I'm not sure why $||L||$ is a spectral norm in this equation even though every other norms are Frobenius norms. Does $||XY||_F\leq ||X||_2||Y||_F$ always hold?