Failure of the Hahn-Banach in ℂₚ

The Hahn-Banach is known to be a crucial tool in functional analysis. The following are lecture notes written a while ago for my functional analysis course on the failure of Hahn-Banach in the [latex]p[/latex]-adic completion of [latex]\bar{\mathbb{Q}}_p[/latex] (denoted by [latex]\mathbb{C}_p[/latex]). For those who are well-versed in the basics of [latex]\mathbb{Q}_p[/latex], I would recommend skipping to around the middle of this page.

Since these are lecture notes, I present as much proof as I can without going into too much gritty detail.

Definition. The [latex]p[/latex]-adic norm of [latex]x\in\mathbb{Q}\setminus \{0\}[/latex] is defined as [latex] |x|_p=p^{-r} [/latex] where [latex]r[/latex] is the unique power of [latex]p[/latex] dividing [latex]x[/latex]. We define [latex]|0|_p:=0[/latex].

Let [latex]x=75/2=5^2\cdot 3\cdot 2^{-1}[/latex]. Then

[latex]
|x|_5=5^2, |x|_3=3, |x|_2=2^{-1}
[/latex]

Definition. A norm [latex]|\cdot|:X\to \mathbb{R}[/latex] is called a non-archimedean norm if

[latex]|x+y|\leq \max\{|x|,|y|\}[/latex]

If a norm does not satisfy the above inequality, then [latex]|\cdot|[/latex] is a archimedean norm.

This inequality is called the strong triangle inequality. As a metric, this norm induces the ultrametric.

Proposition. The [latex]p[/latex]-adic norm is a non-archimedean norm of the field [latex]\mathbb{Q}[/latex].

Proof. By definition, [latex]|x|_p=0\iff x=0[/latex] and [latex]|x|_p\geq 0[/latex] for all [latex]x[/latex]. Since [latex]\mathbb{Z}[/latex] is UFD, [latex]|xy|_p=|x|_p|y|_p[/latex]. Finally, if [latex]x,y\in\mathbb{Q}[/latex],

[latex]
x+y=p^r\frac{u}{v}+p^s\frac{w}{z},\quad\gcd(u,v)=1,\quad\gcd(w,z)=1
[/latex]

Without loss of generality, let [latex]s\geq r[/latex]. Then

[latex]
x+y=p^{r}\frac{uz+p^{s-r}vw}{vz}\implies |x+y|_{p}\leq|x|_p.
[/latex]

where the implication follows since [latex]\gcd(vz,p)=1[/latex] and [latex]\gcd(uz+p^{s-r}vw,p)=1[/latex].
Thus in full generality,

[latex]
|x+y|_p\leq \max\{|x|_p,|y|_p\}\leq |x|_p+|y|_p.
[/latex]

[latex]\blacksquare[/latex]

Definition. The metric completion of [latex]\mathbb{Q}[/latex] with respect to the metric [latex]|\cdot|_p[/latex] is called the [latex]p[/latex]-adic numbers, denoted by [latex]\mathbb{Q}_p[/latex].

Proposition. Any [latex]x\in \mathbb{Q}_p[/latex] has the form

[latex]
x=\sum_{i=k}^\infty a_ip^i
[/latex]

where [latex]a_k\neq 0[/latex] and [latex]a_i\in\{0,…,p-1\}[/latex] for [latex]i>k[/latex].

Consider [latex]\mathbb{Q}_2[/latex] and sums of the form

[latex]
x=\sum_{i=0}^\infty a_ip^i
[/latex]

where [latex]a_i\in\{0,1\}[/latex]. Consider the infinite coordinate [latex]\tilde{x}=(a_0,a_1,a_2,…)[/latex]. Then [latex]\tilde{x}[/latex] is the binary representation of [latex]x[/latex].

Proposition. [latex]\zeta_n\in\mathbb{Q}_p[/latex] if and only if [latex]p\equiv 1\mod n[/latex].

Proof. F.Q. Gouvêa, [latex]p[/latex]-adic Numbers, 1997

[latex]\blacksquare[/latex]

Corollary. [latex]\mathbb{Q}_p[/latex] is not algebraically closed.

Proof. For every [latex]p[/latex], [latex]\zeta_{p+1}\not\in\mathbb{Q}_p[/latex]. Thus [latex]\mathbb{Q}_p[/latex] does not contain the root of the polynomial [latex]x^{p+1}-1=0[/latex].

[latex]\blacksquare[/latex]

Definition. The algebraic closure of [latex]\mathbb{Q}_p[/latex] is denoted by [latex]\overline{\mathbb{Q}}_p[/latex].

Proposition. The norm [latex]|\cdot|_{p}[/latex] extends to [latex]\overline{\mathbb{Q}}_p[/latex].

Proof. Take [latex]x\in\overline{\mathbb{Q}}_p[/latex]. Then [latex]\mathbb{Q}_p(x)[/latex] is a finite extension, of say degree [latex]n[/latex], since [latex]x[/latex] is the root of a polynomial of finite degree. Define [latex]\psi:\overline{\mathbb{Q}}_p(x)\to\overline{\mathbb{Q}}_p(x)[/latex] by

[latex]
w\mapsto xw
[/latex]

Then [latex]\psi[/latex] is [latex]\mathbb{Q}_p[/latex]-linear map, thus define

[latex]
|x|_{p}=\sqrt[n]{|\det \psi|_p}
[/latex]

and check axioms.

[latex]\blacksquare[/latex]

Proposition. [latex]\overline{\mathbb{Q}}_p[/latex] is not complete with respect to [latex]|\cdot|_p[/latex].

Sketch of Proof. Define

[latex]
a_i:=\begin{cases}
\zeta_i,\quad\mbox{if } (i,p)=1,\\ 1,\quad \mbox{otherwise}.
\end{cases}
[/latex]

Define

[latex]
x_{n,m}:=\sum_{i=n}^m a_ip^i
[/latex]

Then [latex]\{x_{n,m}\}[/latex] is Cauchy under the [latex]p[/latex]-norm, but the limit is not in [latex]\overline{\mathbb{Q}}_p[/latex].

[latex]\blacksquare[/latex]

Definition. The metric completion of [latex]\overline{\mathbb{Q}}_p[/latex] with respect to [latex]|\cdot|_p[/latex] is called the [latex]p[/latex]-adic complex numbers, denoted by [latex]\mathbb{C}_p[/latex].

Definition. A metric space [latex]T[/latex] is called spherically complete if for any nested sequence of closed balls [latex]B_1\supseteq B_2\supseteq…[/latex] in [latex]T[/latex],

[latex]
\bigcap_{n\in\mathbb{N}}B_n\neq\emptyset
[/latex]

Some examples of spherically complete fields are [latex]\mathbb{R}[/latex]  and [latex]\mathbb{C}[/latex] by Cantor’s intersection theorem. In fact, these fields are complete with respect to nested sequences of compact sets.

Lemma. [latex]\mathbb{Q}_p[/latex] is spherically complete.

Lemma. [latex]\mathbb{C}_p[/latex] is not spherically complete.

Proof. Since [latex]\mathbb{C}_p[/latex] is separable (See A Course in p-adic Analysis, A. Robert.), there exists a countable dense subset [latex]A[/latex] which can be written as a sequence [latex](a_1,a_2,…)[/latex] where [latex]a_i\in\mathbb{C}_p[/latex]. Fix a sequence [latex](\epsilon_n)_{n=1}^\infty[/latex] where [latex]1>\epsilon_1>\epsilon_2>…>1/2[/latex]. On [latex]\mathbb{C}_p[/latex], define a relation [latex]a\sim b[/latex] iff [latex]|a-b|_p\leq\epsilon_1[/latex]. This is an equivalence relation since

[latex]
a\sim b\sim c\implies |a-c|_p=|a-b+b-c|_p\leq \max\{|a-b|,|b-c|\}\leq \epsilon_1\implies a\sim c
[/latex]

Since the other properties are trivially satisfied, we have our claim. Visually, this equivalence relation partitions [latex]\mathbb{C}_p[/latex] as the set of all [latex]\epsilon_1[/latex] balls where each ball is in fact disjoint! (For [latex]\mathbb{R}[/latex], note that [latex](-1,1)\cap (0,2)=(0,1)[/latex], so the disjointness is indeed very surprising.)

Now observe that [latex]|\cdot|_p[/latex] is a norm mapping to [latex]\mathbb{R}[/latex], but it ONLY takes on values [latex]p^\mathbb{Z}[/latex] when considered under [latex]\mathbb{Q}[/latex]. It can be shown that for [latex]\mathbb{C}_p[/latex], the norm takes on values [latex]p^\mathbb{Q}[/latex]. Moreover, one may notice these values are dense in [latex]\mathbb{R}_+^\times[/latex]. It follows that the diameter of every equivalence class is actually [latex]\epsilon_1[/latex].

We can now iterate. Take [latex]B_1[/latex] to be an equivalency class such that [latex]a_1\notin B_1[/latex]. Then on [latex]B_1[/latex], define an equivalence relation [latex]a\sim b[/latex] iff [latex]|a-b|_p\leq\epsilon_2[/latex]. Then take [latex]B_2\subseteq B_1[/latex] to be an equivalency class such that [latex]a_2\notin B_2[/latex]. Inductively, we have a nested sequence

[latex]
B_1\supseteq B_2\supseteq…\mbox{ such that }\operatorname{diam}(B_i)=\epsilon_i\mbox{ and }a_i\not\in B_i \forall i\in\mathbb{N}
[/latex]

I claim [latex]\cap_{n\in\mathbb{N}} B_n=\emptyset[/latex]. Indeed, if not, then there exists [latex]b\in \cap_{n\in\mathbb{N}} B_n[/latex]. Since [latex]b[/latex] is in each equivalency class, we have [latex]B_i=B_{\epsilon_i}(b)[/latex] for all [latex]i[/latex], so [latex]B_{1/2}(b)\subset\cap_{n\in\mathbb{N}} B_n[/latex] since [latex]\epsilon_i>1/2[/latex]. By construction, no [latex]a_i[/latex] can be contained in [latex]B_{1/2}(b)[/latex], but this contradicts the density of [latex]\overline{\mathbb{Q}}_p[/latex] in [latex]\mathbb{C}_p[/latex].

[latex]\blacksquare[/latex]

Definition. Let [latex]X[/latex] be any set, and let [latex]k[/latex] be a complete field with norm [latex]|\cdot|_k[/latex]. Then we denote by [latex]l^\infty(X)[/latex] the set of all bounded functions on [latex]X[/latex] to [latex]k[/latex]. That is

[latex]
l^\infty(X):=\left\{\varphi:X\to k\mid \sup_{x\in X}|\varphi(x)|_k<\infty\right\}
[/latex]

Lemma. Over [latex]\mathbb{C}_p[/latex], [latex]l^\infty(X)[/latex] is spherically complete and [latex]c_0[/latex] is non-spherically complete.

Proof. Non-Archimedean Functional Analysis, A.C.M. Van Rooij: If [latex]k[/latex] has a dense norm, then [latex]l^\infty[/latex] is spherically complete. Since the norm of [latex]\mathbb{C}_p[/latex] is dense in [latex]\mathbb{R}^+_{\geq 0}[/latex], we are done.

[latex]\blacksquare[/latex]

Recall

Theorem. [Hahn-Banach] Let [latex]V[/latex] be a vector-space over a complete archimedian field. Let [latex]p[/latex] be a sub-linear functional on [latex]V[/latex]. For every linear subspace [latex]E\subseteq V[/latex], for all [latex]f\in E'[/latex] dominated by [latex]p[/latex], there exists a [latex]g\in V'[/latex] such that [latex]f=g|_{E}[/latex] and [latex]||g||\leq ||p||[/latex].

This theorem fails very harshly when the Banach space is non-archimedean; in fact, we can find examples that do not even satisfy the following properties:

Definition. A Banach space [latex]V[/latex] is said to satisfy the Hahn-Banach extension property (HBEP) if for every linear subspace [latex]E\subseteq V[/latex], for all [latex]f\in E'[/latex], there exists a [latex]g\in V'[/latex] such that [latex]f=g|_{E}[/latex] and [latex]||g||=||f||[/latex].

Theorem. HBEP does not hold for Banach spaces over [latex]\mathbb{C}_p[/latex].

Proof. Consider the closed subspace of [latex]l^\infty(\mathbb{N})[/latex]

[latex]
\Delta=\left\{(a,a,…):a\in\mathbb{C}_p\right\}
[/latex]

Consider the linear functional

[latex]
\varphi:\Delta\to\mathbb{C}_p\mbox{ defined by }(a,a,…)\mapsto a
[/latex]

Clearly [latex]||\varphi||=1[/latex]. Now since [latex]\mathbb{C}_p[/latex] is not spherically complete, there is a nested sequence of balls [latex]\{B_{r_i}(x_i)\}[/latex] with empty intersection in [latex]\mathbb{C}_p[/latex]. Consider the nested sequence [latex]B_i^*:=B_{r_i}((x_i,x_i,…))\subseteq l^\infty(\mathbb{N})[/latex]. The intersection of this family is non-empty since [latex]l^\infty(\mathbb{N})[/latex] is spherically complete. Thus let [latex]x[/latex] be in the intersection. Now suppose [latex]\varphi[/latex] extends to a linear map [latex]\psi:l^\infty(\mathbb{N})\to \mathbb{C}_p[/latex] with norm preserved. Then

[latex]
|\psi(x)-x_i|_\infty = |\psi(x)-\varphi(x_i)|_\infty \leq ||\psi||||x-x_i||_{A}=|x-x_i|_{p}\leq\max\{|x|_p,|x_i|_p\}=\max\{r,r_i\}
[/latex]

so [latex]\psi(x)\in\cap_n B_n[/latex], a contradiction.

[latex]\blacksquare[/latex]

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