## mg.metric geometry – Banach fixed point theorem / convergence squeeze

I am currently investigating an iterative learning algorithm and its convergence time. If we let $$x_1 = g(x_0)$$ and let $$epsilon := |x_t – x^*|$$ be our desired error bound from the fixed state, then we have
$$t geq lnleft( frac{epsilon(1-L)}{|g(x_0) – x_0|} right) / ln(L)$$
where L is the Lipschitz constant. My question is this: our function is of the form $$g(x) = frac{1 – s(x)}{Ccdot s(x)}$$ where $$C$$ is some constant and $$s(x)$$ is a function is not always known. If I can verify that the unknown $$s(x)$$ is sandwiched between two polynomials, does this guarantee that $$g(x)$$‘s convergence time can thus be bounded as well? For example if I prove
$$C_1x^{k_1} leq s(x) leq C_2x^{k_2}$$
then can I say
$$text{Conv. time of } C_1x^{k_1} leq text{Conv. time of } s(x) leq text{Conv. time of } C_2x^{k_2}$$

## banach spaces – Biorthogonal weakly null basic sequences

Let $$E$$ be a Banach space, let $$e_{n}in E$$ and $$g_{n}in E^{*}$$ be biorthogonal basic sequences (i.e. $$left=delta_{mn}$$ ). Moreover, both of these sequences are weakly null. (note that existence of these sequences is equivalent to negation of Dunford-Pettis property)

Can we always find $$alpha>0$$, an infinitely dimensional subspace $$Fsubset E$$ and a weakly compact $$Dsubset E^{*}$$, such that $$suplimits_{din D} |left|ge alpha |f|$$, for every $$fin F$$?

(note that this implies reflexivity of $$F$$)

## fa.functional analysis – Estimating certain tensor norms on Banach spaces

Let $$X$$ and $$Y$$ be Banach spaces. An operator $$u:Xto Y$$ s called nuclear if $$u$$ can be written as $$u=sum_{n=1}^infty x_n^*otimes y_n$$ with $$(x_n^*)subseteq X^*$$, $$(y_n)subseteq Y$$ such that $$sum_{n=1}^infty|x_n^*||y_n| Define $$N(u):=inf{sum_{n=1}^infty|x_n^*||y_n|}$$ infimum being taken over all representations. Denote $$C(n):=sup{N(BA):|A|_{ell_1^ntoell_infty^n}leq 1, |B|_{ell_infty^ntoell_infty^n}leq 1}.$$ Is $$suplimits_{ngeq 1}C(n)?

## Universality in the class of separable Banach algebras

Let us consider the class of Banach algebras with homomorphisms that are bounded below but not necessarily isometric.

1. Is there are separable Banach algebra that contains isomorphic images of all separable Banach algebras?

2. Is there a commutative separable Banach algebra that contains commutative separable Banach algebras?

The trick with bounded the distance between commuting projections (of arbitrary norm) does not work in either case.

## rt.representation theory – Under what conditions representations of reductive Lie group in Banach space and in its Garding space have the same length?

Let $$G$$ be a real reductive Lie group (e.g. $$G=GL(n,mathbb{R})$$). Let $$rho$$ be a continuous representation of $$G$$ in a Banach space $$V$$. Let $$V^inftysubset V$$ be the subspace of smooth vectors equipped with the Garding topology. Let $$rho^infty$$ be the natural representation of $$G$$ in $$V^infty$$.

Under what precise technical conditions the representations $$rho$$ and $$rho^infty$$ have the same length?

A reference would be very helpful.

## banach spaces – Finite-dimensional subspaces of \$l_{p}\$ and \$c_{0}\$

Let $$M$$ be a finite-dimensional subspace of $$X=l_{p}$$ or $$c_{0}$$. Let $$epsilon>0$$. Does there exist a projection from $$X$$, of norm $$leq 1+epsilon$$, onto a subspace $$N$$ of $$X$$ with $$Msubseteq N$$ and the Banach-Mazur distance $$textrm{d}(N,l_{p}^{n})$$(resp.$$textrm{d}(N,l_{infty}^{n}))leq 1+epsilon$$, where $$n=textrm{dim}N$$ ?

Thank you!

## banach spaces – Uniform smoothness and twice-differentiability of norms

To get to the simplest case, consider a norm $$|cdot|$$ over $$R^n$$ that is uniformly convex of power-type 2, that is, there is a constant $$C$$ such that $$frac{|x+y| + |x – y|}{2} le 1 + C |y|^2$$ for all $$x$$ with $$|x| = 1$$ and for all $$y$$.

Question: Does this guarantee that $$|cdot|$$ has a second-order Taylor expansion on $$R^n setminus {0}$$, that is, there is a vector $$g$$ and a symmetric matrix $$A$$ such that $$|x + y| = |x| + langle g, y rangle + frac{1}{2} langle Ay, y rangle + o(|y|^2)$$ for all $$x neq 0$$. (Apparently this is a weaker requirement than twice-differentiability of $$|cdot|$$ on $$R^n setminus {0}$$)

It is easy to see that $$|cdot|$$ is differentiable on $$R^n setminus {0}$$, and a classic result of Alexandrov guarantees that the above second-order Taylor expansion holds for any convex function on almost every point $$x$$. It is also known that the norm of any separable Banach space can be approximated arbitrarily well by a power-type 2 norm that is twice differentiable on $$R^n setminus {0}$$ (see Lemma 2.6 here). But I wonder if the original norm itself has a second-order Taylor expansion.

## fa.functional analysis – Wasserstein Space subspace of a Banach Space

Let $$X$$ be a separable and compact (and therefore complete) metric space. Consider the 1-Wasserstein $$P_1(X)$$ space lying over $$X$$ (in the sense that $$xmapsto delta_x$$ is an isometric embedding). Can $$P_1(X)$$ be viewed as a convex subset of some separable Banach space $$E$$? That is, does there exist a separable Banach space $$(E,|cdot|_E)$$ and an isometry (or at-least a bi-Lipschitz map $$phi$$)
$$psi:P_1(X)rightarrow E$$
such that $$psi(P_1)$$ is a convex subset of $$E$$?

I wanted to do something like this with the Arens-Eells space, but the norm is a bit different there…

## fa.functional analysis – A quantity measuring the separability of Banach spaces

Let $$X$$ be a Banach space. It is natural for us to introduce a quantity measuring the separability of sets as follows: for a subset $$A$$ of $$X$$, we set

$$textrm{sep}(A)=inf{epsilon>0: Asubseteq K+epsilon B_{X}$$ for some countable subset $$K$$ of $$X}$$.

Clearly, $$A$$ is separable if and only if $$textrm{sep}(A)=0$$.

It is elementary that a Banach space $$X$$ is separable if $$X^{*}$$ is separable. My question is to give a quantitative version of this known result.

Question. $$textrm{sep}(B_{X})leq Ccdot textrm{sep}(B_{X^{*}})$$ for some universal constant $$C$$ ?

Thank you.

## dg.differential geometry – Reference:Examples of Banach manifolds with function spaces as tangent spaces

I have recently been learning the theory of Banach manifolds through Serge Lang’s book on Differential Manifolds. So far the objects seem rather interesting but my intuition always comes from the finite dimensional case, which admittedly is very helpful but I certainly want to add onto this.

I am wondering if there is a reference where one has examples of Banach manifolds where the tangent spaces are function spaces (ie Sobolev spaces).

In some sense I would like an example that embraces the infinite dimensional tangent space as well as giving me some understanding as to what the topological space/smooth structure should be in the setting.