How I can I prove that the function $F:M(n,\mathbb{R}) \rightarrow M(n,\mathbb{R}):X\mapsto XAX$ for $A \in M(n,\mathbb{R})$ is $C^1$?
I think I have to do it by proving that the function $X\mapsto D_XF$ is continuous, but the thing is this question is part of a bigger question and this is the very first point of the exercise, whilst the second is to calculate its differential, so it doesn't make sense for me to calculate its differential here. Maybe I can somehow upperboud the norm operator of $D_XF$ without calculating it?
However, I was able to calculate that $D_XF:H\mapsto XAH+HAX$