# About finite representability of Banach space

Can someone please tell me the brief sketch (or any known reference) of the following results?

1. Why $$\ell_2$$ is finitely representable in any infinite-dimensional Banach space?
2. Why every Banach space is finitely representable in $$c_0$$?

A Banach space $$Y$$ is said to be finitely representable in some Banach space $$X$$ if for any finite dimensional subspace $$F$$ of $$Y$$ and $$\varepsilon>0$$ there exists an isomorphism $$T:F\to X$$ such that $$\|T\|\|T^{-1}\|\leq 1+\varepsilon$$.

• Yes, of course. Thanks for pointing out. – Tanmoy Paul Oct 9 '18 at 19:50
• You might want to read about Dvoretzky's theorem – Matthew Daws Oct 9 '18 at 20:01

1 is very hard and 2 is very easy. For 2, take an $$\varepsilon$$-net $$(x_i)_{i=1}^m$$ in the sphere of the $$n$$ dimensional space ($$m$$ depends on $$n$$ and $$\varepsilon$$), and norming functionals $$f_i$$'s. Now check that the map from the $$n$$-dimensional space into $$\ell_{\infty}^m\subset c_0$$ given by $$x\to (f_1(x), f_2(x), \ldots, f_m(x))$$ works.
For 1 there is a bit of easier proof if you don't care about the constant being $$1+\varepsilon$$ and if you are familiar with spreading models but still pretty involved. But standard reference for 1 is Milman-Schechtman's book
The first one is Dvoretzky's theorem (or, more precisely, a consequence of Dvoretzky's theorem), which says not only that you can find $$1+\epsilon$$ isomorphic copies of $$\ell_2^n$$ in infinite dimensional spaces, but that you can find a $$1+\epsilon$$ copy of $$\ell_2^n$$ in any spaces whose dimension is at least $$N$$, where $$N$$ is a function of $$\epsilon$$ and $$n$$. The proof is technical, but the idea is simple. We use the idea of concentration of measure. We consider the norm $$\|\cdot\|$$ on the $$N$$-dimensional space $$\mathbb{R}^N$$. By John's theorem, after applying a suitable invertible linear translation, we can assume the Lipschitz constant of $$\|\cdot\|$$ with respect to the $$\ell_2^n$$ norm of $$\mathbb{R}^N$$ is at most $$N^{1/2}$$. Then one can show that, since the function $$\|\cdot\|$$ is concentrated around its mean (or median, depending on which version of concentration you use), there is a very small measure exceptional set of the sphere such that on the complement of the exceptional set, the norm $$\|\cdot\|$$ is almost exactly a fixed multiple of the $$\ell_2^N$$ norm. This exceptional set is small enough to be able to find an $$n$$-dimensional subspace which contains none of the exceptional set, and therefore the $$\ell_2$$ and $$\|\cdot\|$$ norms are almost exactly a multiple of each other on this $$n$$-dimensional subset.
Alternatively, if you know Krivine's theorem (which I personally find easier to understand), then you can deduce Dvoretzky's theorem. Every infinite dimensional Banach space has a basic sequence. There exists $$1\leqslant p\leqslant \infty$$ such that we can find $$1+\epsilon$$ $$\ell_p^n$$ spaces in the sequence (and even as blocks of the sequence). If $$p=\infty$$, this means $$c_0$$ (and by your second question, every space) is finitely representable in your space, including $$\ell_2$$. If $$1\leqslant p<\infty$$, this means $$\ell_p$$ is finitely representable in your space. But since $$L_p$$ is finitely representable in $$\ell_p$$, this means $$L_p$$ is finitely representable in your space. And $$L_p$$ contains an isometric copy of $$\ell_2$$ as the closed span of independent gaussian random variables. This is a roundabout way, but it contains a lot of interesting pieces along the way.