David R. MacIver's Blog

Generalizing local approximations to global ones

In a previous blog post I looked at the question “How well can you approximate a set of bounded real-valued functions by a single continuous one?”.

This post looks at the generalization that’s at the heart of what I’m writing up: How does this work when we replace \(\mathbb{R}\) with an arbitrary normed space?

I won’t be looking at the question of whether a center exists (it turns out that the question is better answered independently of this theory). Instead we will generalize the result that $$\frac{1}{2} \sup\limits_x (\mathcal{F}^*(x) – \mathcal{F}_*(x)) \leq r(\mathcal{F}, C(X))$$ and that this inequality was an equality when \(X\) was normal.

The functions \(\mathcal{F}^*(x)\) and \(\mathcal{F}_*(x)\) don’t really make sense for arbitrary vector spaces (they depend inherently on the ordered structure of \(\mathbb{R}\)), but their difference does. The following lemma will help point out how to generalize them:

Theorem: Define the oscillation of a family \(\mathcal{F}\) at a point \(x\) as $$\omega_x(\mathcal{F}) = \inf\{ \mathrm{diam}(\mathcal{F}(U)) : U \textrm{ open }, U \ni x \} $$

where \(\mathcal{F}(A) = \bigcup\limits_{f \in \mathcal{F}} f(A) \). (This is a natural generalization of the usual definition).

Then $$ \mathcal{F}^*(x) - \mathcal{F}_*(x) = \omega_x(\mathcal{F})$$

Proof:

Let \(\epsilon > 0\).

Let \(U \ni x\) such hat \(\mathrm{diam}(\mathcal{F}(U)) \leq \omega_x(\mathcal{F}) + \epsilon\). Then for \(u, v \in U\) and \(f, g \in \mathcal{F}\) we have \(|g(x) - f(u)| \leq \omega_x(\mathcal{F}) + \epsilon\). So \(f(u) \leq g(v) + \omega_x(\mathcal{F}) + \epsilon\). Taking the supremum and infimum we have
$$\mathcal{F}^*(x) \leq \sup \mathcal{F}(u) \leq \inf \mathcal{F}(v) + \omega_x(\mathcal{F}) + \epsilon \leq \mathcal{F}_*(x) + \omega_x(\mathcal{F})(x) + \epsilon$$

Hence $$\mathcal{F}^*(x) - \mathcal{F}_*(x) \leq \omega_x(\mathcal{F})(x) + \epsilon$$

Letting \(\epsilon \to 0\) we get $$\mathcal{F}^*(x) - \mathcal{F}_*(x) \leq \omega_x(\mathcal{F})(x)$$

The other direction of the inequality follows similarly (because in \(\mathbb{R}\), \(\mathrm{diam}(A) = \sup A - \inf A \).
QED

In particular $$r(\mathcal{F}, C(X)) \geq \sup\limits_x \omega_x(\mathcal{F})$$, with equality if \(X\) is normal.

This theorem suggests an obvious generalization: When is it the case that \(d(\mathcal{F}, C(X, V)) = \sup\limits_x \omega_x(\mathcal{F})\)?

It turns out the answer is “rarely”, but mostly because this is the wrong generalization. The fact that the oscillation so well defines the approximability for the real case is because we can find good centers - in \(\mathbb{R}\) we have \(\mathrm{diam}(A) = 2 r(A)\). In general this relation does not hold. This leads us to define the following modification:

Define the radial oscillation of a family \(\mathcal{F}\) at a point \(x\) as $$\rho_x(\mathcal{F}) = \inf\{ r(\mathcal{F}(U)) : U \textrm{ open }, U \ni x \} $$

Write \(\rho(\mathcal{F}) = \sup\limits_x \rho_x(\mathcal{F})\).

The following theorem is fairly immediate from the definition:

Theorem: $$\rho(\mathcal{F}) \leq r(\mathcal{F}, C(X, V))$$

Proof:

Let \(\mathcal{F} \subseteq B(g, R)\)

Let \(x \in X\). Pick \(U \ni x\) such that for \(u \in U\), \(||g(u) - g(x)|| < \epsilon\).

Then for \(u \in U\) and \(f \in \mathcal{F}\) we have \(||f(u) - g(x)|| \leq R + \epsilon\).

Therefore letting \(\epsilon \to 0\) we have \(\rho_x(\mathcal{F}) \leq R\). Taking the infimum over possible values of R we have the desired result. QED

So the question is this: For what pairs X, V is this inequality an equality? We saw that if \(V = \mathbb{R}\) then X being normal was sufficient. Is normal generally sufficient?

We will say that X is OCA(V) (“has optimal continuous approximations into V”) if this is an equality for every bounded family \(\mathcal{F}\).

Advance warning: Unfortunately we were never able to resolve the full answer to this. We have some quite nice sufficient conditions, and a bunch of partial necessary conditions, but we were never able to get a general characterization that satisfied us. Still, I think some of the partial results we got are quite interesting.

That depressing note aside, here’s one of our main sufficiency conditions:

Theorem: Let \(\kappa\) be a cardinal (including a finite one), \(X\) a topological space with the property that every open cover of fewer than \(\kappa\) open sets has a subordinate partition of unity and V a normed space such that every bounded subset of V has an \(\epsilon\)-net of size \(< \kappa\) for any \(\epsilon > 0\). Then X is OCA(V).

This has a number of immediate corollaries.

Corollary: If \(X\) is paracompact then \(X\) is OCA(V) for any V.
Corollary: If \(X\) is countably paracompact then \(X\) is OCA(V) for any separable V.
Corollary: If \(X\) is normal then \(X\) is OCA(V) for any finite dimensional V.

In order to prove the result we will need the following lemma:

Lemma: Let \(U_\alpha\) be an open cover of X such that \(\mathcal{F}(U_\alpha) \subseteq \overline{B}(v_\alpha, R)\). Let \(p_\alpha\) be a partition of unity subordinate to \(U_\alpha\). Let $$g(x) = \sum\limits_\alpha p_\alpha(x) v_\alpha$$

Then \(\mathcal{F} \subseteq \overline{B}(g, R)\).

Proof:

Let \(x \in X\). Then \(g(x)\) is a convex combination of \(v_\alpha\) such that \(f(x) \in \overline{B}(v_\alpha, R)\). Thus \(v_\alpha \in \overline{B}(x, R)\). Balls are convex, hence \(g(x) \in \overline{B}(x, R))\). QED

We will now prove the theorem:

Let \(\epsilon > 0\) and let \(\{u_\alpha : \alpha \in A \}\) be an \(\epsilon\)-net with \(|A| < \kappa\).

By the definition of \(rho\) for every \(x \in X\) we may find an open set \(V_x\) such that \(\mathcal{F}(V_x) \subseteq \overline{B}(c, \rho(\mathcal{F}) + \epsilon)\). But then we have \(||c - v_\alpha|| \leq \epsilon\) for some \(v_\alpha\), so in fact \(\mathcal{F}(V_x) \subseteq \overline{B}(v_\alpha, \rho(\mathcal{F}) + 2\epsilon)\)

Now let $$U_\alpha = \bigcup \{ V_x : \mathcal{F}(V_x) \subseteq \overline{B}(v_\alpha, \rho(\mathcal{F}) + 2\epsilon) \}$$

We have \(\mathcal{F}(U_\alpha) \subseteq \overline{B}(v_\alpha, \rho(\mathcal{F}) + 2\epsilon)\). Further because \(|A| < \kappa\), by hypothesis we can find a subordinate partition of unity \(p_\alpha\). We now apply our lemma and get a continuous function g such that \(\mathcal{F} \subseteq B(g, \rho(\mathcal{F}) + 2 \epsilon)\). Hence \(r(\mathcal{F}, C(X, V)) \leq \rho(\mathcal{F}) + 2\epsilon\). Letting \(\epsilon \to 0\) the result is proved. QED

Notes

• While the exact form of this theorem is probably novel, none of the details are. Unfortunately I can no longer find my notes on the relevant literature, as we arrived at this ourselves and then found the existing work later.
• The original form of this particular proof used a continuous selection theorem from A unified theorem on continuous selections by Michael and Pixley. We ended up unpicking the proof and proving it directly in order to get the tight cardinal bounds.
• This is in some sense our “most general” result about OCA. There will be a few other theorems which give some examples not covered by this, but this covers far more cases than others
• One natural question that emerges is to what extent is the converse result true. In particular if X is OCA(V) for every V, is X paracompact? We have some partial results of this form (which I’ll cover in a later post), but we never managed to get a full converse. As far as I know it’s an open problem

Comments

David R. MacIver on 2012-03-11 17:35:23:

[...] post is about two partial converse results to the main theorem from my last post. Both of these results were established by John rather than [...]