Equivalence of $R^m$ and $l^2$(square-summable sequence) in the proof of Grothendieck's inequality

Question

This is from High dimensional probability by Vershynin.

In the proof of Grothendieck's inequality, for $u,v \in \mathbb{R}^n$, we find functions $\phi, \varphi \colon \mathbb{R} \to l^2$ such that $\frac{2}{\pi}\arcsin<\phi(u), \varphi(v)> = \beta<u,v>$(where $l^2$ is the Hilbert space of square-summable sequences).

We'd like to use the identity $\frac{2}{\pi} \arcsin <u,v> = E sign<g,u> sign<g,v>$, where $u,v \in \mathbb{R}^n$, and $g \sim N(0,I_n)$. However, $\phi(u)$ lives in $l^2$ so we can't use this identity(I believe).

The book says that without loss of generality, we may assume the Hilbert space of the co-domain of $\phi$ and $\varphi$ to be $\mathbb{R}^M$ for some $M \in \mathbb{N}$. But how can we make this assumption when $l^2$ is infinite-dimensional, and $\mathbb{R}^m$ is not?

I'm just wondering why we can make this assumption.

Thank you.

And here's what the book means by "again".

Answer 1

The goal is to prove there is a universal constant $K$ such that for any $m$ and $n$, and any $m+n$ unit vectors $u_1,\dots,u_m,v_1,\dots,v_n$ in a (potentially abstract) Hilbert space $(H,\langle\cdot,\cdot\rangle_H)$, $$ |\sum_{i,j}a_{ij}\langle u_i, v_j\rangle_H|\le K. $$ We claim it suffices to consider only any $m+n$ unit vectors in the concrete Hilbert space $\mathbb R^N$ with its usual inner product $\langle\cdot,\cdot\rangle_{\mathrm{Euc}}$ for some $N\le m+n$. To see it, we consider the following.

Given any $m+n$ vectors $u_1,\dots,u_m,v_1,\dots,v_n$ in a potentially abstract Hilbert space $(H,\langle\cdot,\cdot\rangle_H)$, let $W$ be the span of $u_1,\dots,u_m,v_1,\dots,v_n$ in $H$. The sub-Hilbert space $(W,\langle\cdot,\cdot\rangle_H|_{W\times W})$ is a finite-dimensional real vector space of dimension $N\le m+n$, and as such it can be isometrically identified with $\mathbb R^N$ by making an arbitrary choice of orthonormal basis $w_1,\dots,w_N$ for $W$ and mapping it to the standard orthonormal basis $e_1,\dots,e_N$ for $\mathbb R^N$ with its usual Euclidean inner product $\langle\cdot,\cdot\rangle_{\mathrm{Euc}}$. Let $T\colon W\to \mathbb R^N$ be this isomorphism.

Then note that the quantity we care about $|\sum_{i,j}a_{ij}\langle u_i, v_j\rangle_H|$ is simply equal to $$ |\sum_{i,j}a_{ij}\langle T(u_i), T(v_j)\rangle_{\mathrm{Euc}}|, $$ and if we show that this is bounded by some universal constant $K$, then we may as well just assume from the start that $(H,\langle\cdot,\cdot\rangle_H) = (\mathbb R^N,\langle\cdot,\cdot\rangle_{\mathrm{Euc}})$ for some $N$.