This question is related to my question [Can we choose an element from a class?](https://mathoverflow.net/questions/387353/can-we-choose-an-element-from-a-class).
However, I decided to create a separate question.


Let $H$ be a complex Hilbert space and $H\_1,\dotsc,H\_n$ be closed subspaces of $H$.
Set $H\_0\mathrel{:=}H\_1\cap H\_2\cap\dotsb\cap H\_n$ and let
$P\_i$ be the orthogonal projection onto $H\_i$, $i=0,1,2,\dotsc,n$.
I study the functions $f\_n:[0,1]\to\mathbb{R}$ defined by
$$
f\_n(c)=\sup\{\lVert P\_n\dotsm P\_2 P\_1-P\_0\rVert \mathrel|
c\_F(H\_1,\dotsc,H\_n)\leqslant c\},\,c\in[0,1],
$$
where the supremum is taken over all complex Hilbert spaces $H$ and
systems of closed subspaces $H\_1,\dotsc,H\_n$ of $H$
for which the Friedrichs number $c\_F(H\_1,\dotsc,H\_n)$ is less than or equal to $c$
(the Friedrichs number is a certain numerical characteristic of a system of subspaces).
Note that all such systems of subspaces **do not** form a set.
Despite this, the function $f\_n$ is well-defined
(see my Question [Can we take a supremum over all Hilbert spaces?](https://mathoverflow.net/questions/375759/can-we-take-a-supremum-over-all-hilbert-spaces)).
Indeed, let $A\_{n}(c)$ be the set of all $a\in\mathbb{R}$ for which there exist a complex Hilbert space $H$ and a system of closed subspaces $H\_1,\dotsc,H\_n$ of $H$ such that
$c\_F(H\_1,\dotsc,H\_n)\leqslant c$ and $\lVert P\_n\dotsm P\_2 P\_1-P\_0\rVert=a$.
Then by the axiom (scheme) of separation $A\_{n}(c)$ is a **set** and thus we can take its supremum.


I need to show that $f\_n(c)\leqslant g\_n(c)$ for some function $g\_n$.
I argue as follows.
Consider arbitrary element $a\in A\_{n}(c)$.
Then there exist a complex Hilbert space $H$ and a system of closed subspaces $H\_1,\dotsc,H\_n$ of $H$
such that $c\_F(H\_1,\dotsc,H\_n)\leqslant c$ and $\lVert P\_n\dotsm P\_2 P\_1-P\_0\rVert=a$.
After this I work with this system of subspaces $(H;H\_1,\dotsc,H\_n)$ and show that
$\|P\_n\dotsm P\_2 P\_1-P\_0\|\leqslant g\_n(c)$.
Thus $a\leqslant g\_n(c)$.
Since this inequality holds for every $a\in A\_{n}(c)$, we conclude that
$\sup A\_{n}(c)\leqslant g\_n(c)$, i.e., $f\_n(c)\leqslant g\_n(c)$.


**Questions.** Are all these arguments correct, say, in the axiomatic theory ZFC?
Essentially, the core of my worries is the following.
Unfortunately, I do not understand if the **function** $A\_n(c)\ni a\mapsto (H;H\_1,...,H\_n)$
such that $c\_F(H\_1,\dotsc,H\_n)\leqslant c$ and $\lVert P\_n\dotsm P\_2 P\_1-P\_0\rVert=a$,
is needed in the arguments above or not.
If a function is needed here, it turns out that the "Axiom of Choice for classes" is needed here,
and I do not know what to do with this.
On the one hand, it seems that the function is not needed here.
On the other hand, we choose a system of subspaces for each $a\in A\_n(c)$;
the set $A\_n(c)$ can be infinite and we need to consider **all** $a\in A\_n(c)$.
Therefore, perhaps, a function is needed here.
**Explain to me, please, whether a function is needed here or not?**


Please help me.



Is a function needed here?
The answers are as follows: these arguments are correct, the indicated function is not needed.


Indeed, your argument might as well prove that for all $H\_1,\dotsc,H\_n$, and $H$ such that $c\_F(H\_1,\dotsc,H\_n)\le c$, you have $\lVert P\_n\dotsm P\_2P\_1−P\_0\rVert \le g\_n(c)$.
Quantifying over a collection of sets is perfectly legitimate even if this collection of sets forms a proper class.
There is no need to mention the set $A\_n(c)$ at all, it only creates further confusion.


