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I am writing a paper in which I give the following definition.

Let $n$ be a natural number greater than or equal to $2$. Define a function $f_n:[0,1]\to\mathbb{R}$ by $$ f_n(c)=\sup\{\|P_n...P_2 P_1-P_0\|\,|\,c_F(H;H_1,...,H_n)\leqslant c\}, $$ where the supremum is taken over all complex Hilbert spaces $H$ and systems of closed subspaces $H_1,...,H_n$ of $H$ for which the Friedrichs number $c_F(H;H_1,...,H_n)$ is less than or equal to $c$, $P_i$ denotes the orthogonal projection onto $H_i$ in $H$, $i=1,2,...,n$, and $P_0$ denotes the orthogonal projection onto the subspace $H_0:=H_1\cap H_2\cap...\cap H_n$ in $H$, $c\in[0,1]$.

Is the notation in this definition clear and correct? Or do I have to write something like the following?

For a Hilbert space $M$ and a closed subspace $N$ of $M$ denote by $P^M_N$ the orthogonal projection onto $N$ in $M$. Let $n$ be a natural number greater than or equal to $2$. Define a function $f_n:[0,1]\to\mathbb{R}$ by $$ f_n(c)=\sup\{\|P^H_{H_n}...P^H_{H_2} P^H_{H_1}-P^H_{H_1\cap H_2\cap...\cap H_n}\|\,|\,c_F(H;H_1,...,H_n)\leqslant c\}, $$ where the supremum is taken over all complex Hilbert spaces $H$ and systems of closed subspaces $H_1,...,H_n$ of $H$ for which the Friedrichs number $c_F(H;H_1,...,H_n)$ is less than or equal to $c$, $c\in[0,1]$.

I will be very grateful for any comments and answers.

Ivan Feshchenko
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    Don't write all those subscripts and superscripts. They don't add anything and you still need to explain what $P_{H_i}^H$ means anyway. I would make the dependence on $H$ explicit only if I had to work with those projections for multiple $H$ simultaneously. – plop Jun 08 '21 at 13:05

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