{"id":16285,"date":"2022-10-10T16:57:21","date_gmt":"2022-10-10T14:57:21","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=16285"},"modified":"2022-11-26T22:39:55","modified_gmt":"2022-11-26T21:39:55","slug":"little-ell-p","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2022\/10\/10\/little-ell-p\/","title":{"rendered":"Little ell p"},"content":{"rendered":"
\"Stefan<\/a>
Stefan Banach (1892 - 1945)<\/figcaption><\/figure>\n

I teach topology and differential calculus this semester. This is the occasion to play with many nice mathematical concepts, including one of my favorite Banach spaces : $\\ell^p$ spaces. These spaces are at the same time simple, important, and subtle. This tiny post collects some basic properties of these spaces, just for pleasure. I will improve this tiny post from time to time.<\/p>\n

In what follows, $\\mathbb{K}\\in\\{\\mathbb{R},\\mathbb{C}\\}$.<\/p>\n

$\\ell^p$ spaces.<\/strong> For all $p\\in[1,\\infty)$, $\\ell^p=\\ell^p(\\mathbb{N},\\mathbb{K})$ is the set of sequences $(x_n)_{n\\in\\mathbb{N}}$ in $\\mathbb{K}$ such that $$\\|x\\|_p:=\\sum_n|x_n|^p<\\infty.$$ We also define $\\ell^\\infty=\\ell^\\infty(\\mathbb{N},\\mathbb{K})$ the set of sequences $(x_n)_{n\\in\\mathbb{N}}$ in $\\mathbb{K}$ such that $$\\|x\\|_\\infty:=\\sup_n|x_n|<\\infty.$$ These spaces naturally generalize $\\mathbb{R}^n$ to $\\mathbb{R}^\\infty$. Note by the way that $[0,1]^\\infty\\subset\\ell^p$, $p\\in[1,\\infty]$, but the topology induced by $\\ell^p$ on $[0,1]^\\infty$ is not the product topology, it is stronger, in particular for $p=\\infty$ it corresponds to uniform convergence rather than convergence of each coordinate.<\/p>\n

We always have $\\ell^{p_1}\\subsetneq\\ell^{p_2}$ for all $1\\leq p_1<p_2\\leq\\infty$.<\/p>\n

Three spaces are of special importance: $\\ell^1$, $\\ell^2$, and $\\ell^\\infty$.<\/p>\n

Functional point of view.<\/strong> A sequence $(x_n)_{n\\in\\mathbb{N}}$ in $\\mathbb{K}$ is a function $\\mathbb{N}\\to\\mathbb{K}$, $x(n):=x_n$. If we equip $\\mathbb{N}$ with the discrete topology, the discrete $\\sigma$-field, and the counting measure $\\mathrm{d}n$, then $$\\ell^p(\\mathbb{N},\\mathbb{K})=L^p(\\mathbb{N},\\mathrm{d}n,\\mathbb{K})\\quad\\text{and}\\quad \\|x\\|_p^p=\\int |x(n)|^p\\mathrm{d}n,$$ while $\\ell^\\infty=L^\\infty(\\mathbb{N},\\mathrm{d}n,\\mathbb{K})=\\mathcal{C}_b(\\mathbb{N},\\mathbb{K})$. But let us study these spaces from scratch.<\/p>\n

For all $n\\in\\mathbb{N}$, we set $e_n:=\\mathbf{1}_n$, in such a way that for all $x\\in\\ell^p$, $x=\\sum_nx_ne_n$.<\/p>\n

Completeness.<\/strong> The spaces $\\ell^p$, $p\\in[1,\\infty]$, are Banach spaces : complete normed vector spaces. The vector space nature and the norm axioms are not difficult to check. To establish the completeness of $\\ell^p$ when $p\\in[1,\\infty)$, we consider a Cauchy sequence $(x^{(m)})$ in $\\ell^p$. For all $n$, $(x^{(m)}_n)$ is a Cauchy sequence in $\\mathbb{R}^n$ which is complete, hence we get a sequence $x^*=(x^*_n)$ such that $x^{(m)}_n\\to x^*_n$ as $m\\to\\infty$, for all $n$. Next, since $(x^{(m)})$ is bounded in $\\ell^p$, for all $N$, $$\\sum_{n=0}^N|x^*_n|^p=\\lim_{m\\to\\infty}\\sum_{n=0}^N|x^{(m)}_k|^p\\leq\\sup_m\\|x^{(m)}\\|_p^p<\\infty$$ thus $x^*\\in\\ell^p$ by taking the limit $N\\to\\infty$. Next, $$\\sum_{n=0}^N|x^{(m)}_n-x^*_n|^p=\\lim_{k\\to\\infty}\\sum_{n=0}^K|x^{(m)}_n-x^{(k)}_n|^p\\leq\\varlimsup_{k\\to\\infty}\\|x^{(m)}-x^{(k)}\\|_p^p,$$ which implies that $x^{(m)}\\to x^*$ in $\\ell^p$ since $(x^{(m)})$ is Cauchy in $\\ell^p$. This also works for $\\ell^\\infty$.<\/p>\n

For $p=2$, we get a Hilbert space $\\ell^2$ with dot product $x\\cdot y:=\\sum_nx_n\\overline{y_n}$, while when $p\\neq2$, the norm $\\left\\|\\cdot\\right\\|_p$ does not satisfy the parallelogram identity and is thus not Hilbertian.<\/p>\n

Not locally compact.<\/strong> The argument can be adapted to any infinite dimensional normed vector space. Namely, for $\\ell^p$, $p\\in[1,\\infty]$, for any $r>0$, $x_n:=re_n=r\\mathbf{1}_n$ satisfies $\\|x_n\\|_p=r$ and $\\|x_n-x_m\\|=r2^{1\/p}$ if $n\\neq m$, also for any $\\varepsilon>0$, if $\\overline{B}(0,r)$ was coverable by a finite number of balls of radius $\\varepsilon>0$, then one of these ball would contain at least two distinct points $x_n$ and $x_m$, and this would give $r2^{1\/p}=\\|x_n-x_m\\|\\leq2\\varepsilon$, which is impossible when $\\varepsilon<r2^{1\/p-1}$.<\/p>\n

Non separability of $\\ell^\\infty$.<\/strong> For $I\\subset\\mathbb{N}$, set $e_I:=\\sum_{i\\in I}e_i$. Then for $I\\cap J$, we have $\\|e_I-e_J\\|_\\infty=1$, thus $B(e_I,\\tfrac{1}{2})\\cap B(e_J,\\tfrac{1}{2})=\\varnothing$, thus $\\ell^\\infty$ can be covered by an uncountable family of disjoint non-empty balls, since the set of non-empty subsets of $\\mathbb{N}$ is uncountable. This argument does not work for $\\ell^p$ with $p\\in[1,\\infty)$, since in this case $e_I\\in\\ell^p$ imposes that $I$ is finite, and the set of non-empty finite subsets of $\\mathbb{N}$ is countable.<\/p>\n

Separability of $\\ell^p$, $p\\in[1,\\infty)$.<\/strong> If suffices to consider $A:=\\cup_nA_n$ where $$A_n:=\\Bigr\\{\\sum_{i=0}^nq_ie_i:q\\in\\mathbb{Q}_{\\mathbb{K}}^{n+1}\\Bigr\\}.$$ Hence for all $x\\in\\ell^p$ and all $\\varepsilon>0$, denoting $\\pi_n(x):=\\sum_{i=0}^{n-1}x_ie_i$, we have $\\|x-\\pi_n(x)\\|_p\\leq\\varepsilon$ for a large enough $n$, and then $\\|\\pi_n(x)-y_n\\|_p\\leq\\varepsilon$ for some $y\\in A_n$ thanks to the density of $\\mathbb{Q}_{\\mathbb{K}}$ in $\\mathbb{K}$. Note that the approximation $\\|x-\\pi_n(x)\\|_p\\leq\\varepsilon$ of $x$ by the finitely supported sequence $\\pi_n(x)$ does not work in $\\ell^\\infty$. Actually the closure in $\\ell^\\infty$ of the set of finitely supported sequences is the set of sequences which tend to $0$ at $\\infty$, denoted $\\ell^\\infty_0$, which is strictly smaller than $\\ell^\\infty$, and which is a separable Banach subspace of $\\ell^\\infty$.<\/p>\n

H\u00f6lder inequality.<\/strong> For all $p\\in[1,\\infty]$, $q:=p\/(p-1)\\in[1,\\infty]$, all $x\\in\\ell^p$, $y\\in\\ell^q$, $$\\sum_n|x_ny_n|\\leq\\|x\\|_p\\|y\\|_q.$$ It follows from the H\u00f6lder inequality on $\\mathbb{R}^n$ for $\\pi_n(x)=\\sum_{i=0}^{n-1}x_ie_i$ and $\\pi_n(y)=\\sum_{i=0}^{n-1}y_ie_i$. The equality is achieved when $|x_n|^p$ and $|y_n|^q$ are proportional (possibly asymptotically).<\/p>\n

Dual of $\\ell^p$, $p\\in[0,\\infty)$.<\/strong> Recall that the dual of a normed vector space $X$ is the normed vector space $X'=L(X,\\mathbb{K})$ of continuous linear forms, namely continuous linear mappings $X\\to\\mathbb{K}$.<\/p>\n

For all $p\\in[1,\\infty)$, denoting $q:=1\/(1-1\/p)=p\/(p-1)\\in(1,\\infty]$ the H\u00f6lder conjugate of $p$, the map $\\Phi:\\ell^q\\mapsto(\\ell^p)'$ defined for all $y\\in\\ell^q$ and $x\\in\\ell^p$ by $$\\Phi(y)(x):=\\sum_nx_ny_n$$ is a bijective linear isometry. In particular $$(\\ell^p)'\\equiv\\ell^q.$$ In particular $(\\ell^2)'\\equiv\\ell^2$, $(\\ell^1)'\\equiv\\ell^\\infty$, and $((\\ell^p)')'\\equiv\\ell^p$ (reflexivity) for $p\\in(1,\\infty)$.<\/p>\n

Proof.<\/strong> The map $\\Phi$ is well defined thanks to the H\u00f6lder inequality, and $\\|\\Phi(x)\\|_{(\\ell^p)'}\\leq\\|x\\|_q$. Let us establish the equality, hence it is injective or into, and then that it is surjective or onto.<\/p>\n

Let us consider the case $p=1$ ($q=\\infty$). Let $y\\in\\ell^\\infty$. There exists $(n_k)$ such that $|y_{n_k}|\\to\\|y\\|_\\infty$ when $k\\to\\infty$. For all $k$, $x_k:=\\tfrac{|y_{n_k}|}{y_{n_k}}e_{n_k}\\in\\ell^1$ satifies $\\|x_k\\|_1=1$ and $\\Phi(y)(x_k)=|y_{n_k}|\\to\\|y\\|_\\infty$ when $k\\to\\infty$, hence $\\|\\Phi\\|=\\|y\\|_\\infty$, and $\\Phi$ is an isometry and thus an injection. For the surjectivity, if h$\\varphi\\in(\\ell^1)'$ then for all $x\\in\\ell^1$, $\\varphi(x)=\\sum_ny_nx_n$ with $y_n:=\\varphi(e_n)$, but $|y_n|\\leq\\|\\varphi\\|\\|e_n\\|_1=\\|\\varphi\\|<\\infty$ hence $y\\in\\ell^\\infty$ and $\\varphi=\\Phi(y)$.<\/p>\n

Let us consider the case $p>1$ ($q<\\infty$). The equality case in the H\u00f6lder inequality $x_n:=\\tfrac{y_n}{|y_n|}|y_n|^{q-1}$ satisfies $\\|x\\|_p^p=\\|y\\|_q^q$ since $p(q-1)=q$, hence $\\Phi(y)(x)=\\|y\\|_q^q=\\|y\\|_q\\|x\\|_p$, hence $\\|\\Phi(y)\\|_{(\\ell^p)'}=\\|y\\|_q$, and $\\Phi$ is an isometry hence it is injective. Let us show that $\\Phi$ is surjective. Let $\\varphi\\in(\\ell^p)'$, then for all $x\\in\\ell^p$, $\\varphi(x)=\\sum_nx_ny_n$ where $y_n:=\\varphi(e_n)$ with $e_n:=\\mathbf{1}_n$. Suppose by contradiction that $y\\not\\in\\ell^q$. Denoting $\\pi_N(z):=(z_0,z_1,\\ldots,z_{N-1},0,0,\\ldots)$, and $x_n:=\\frac{|y_n|}{y_n}|y_n|^{q-1}$, since $y\\not\\in\\ell^q$,
\n$$
\n\\frac{\\Phi(y)(\\pi_N(x))}{\\|\\pi_N(x)\\|_p}
\n=\\|\\pi_N(y)\\|_q\\underset{n\\to\\infty}{\\longrightarrow}\\infty,
\n\\quad\\text{which contradicts $\\Phi(y)\\in(\\ell^p)'$}.
\n$$<\/p>\n

Dual of $\\ell^\\infty$.<\/strong> The dual of $\\ell^\\infty=(\\ell^1)'$ is strictly larger<\/strong> than $\\ell^1$. More precisely, the map $\\Phi:\\ell^1\\mapsto(\\ell^\\infty)'$ defined by $\\Phi(y)(x):=\\sum_nx_ny_n$ is a linear isometry which is injective but not surjective. In other words, $\\ell^1\\subsetneq(\\ell^\\infty)'=((\\ell^1)')'$<\/strong>, in other words $\\ell^1$ is not reflexive<\/strong>.<\/p>\n

Proof.<\/strong> The isometry (and thus the injectivity) comes from the H\u00f6lder inequality
\n$|\\Phi(y)(x)|\\leq\\|x\\|_\\infty\\|y\\|_1$ and its equality case $x_n=\\tfrac{|y_n|}{y_n}$, which gives\u00a0 $\\|\\Phi(y)\\|=\\|y\\|_1$. It remains to establish that $\\Phi$ is not surjective. Consider the following subspace of $\\ell^\\infty$:
\n\\[
\nS:=\\Bigr\\{(x_n):x_*:=\\lim_{n\\to\\infty}x_n\\text{ exists}\\Bigr\\}.
\n\\]
\nThe linear functional $x\\mapsto x_*$ is bounded has unit norm on $S$. Thanks to the Hahn--Banach (for the non separable Banach space $\\ell^\\infty$) we can extend it into $L:\\ell^\\infty\\to\\mathbb{K}$ in such a way that $|Lx|\\leq\\|x\\|_\\infty=\\sup_n|x_n|$ and $Lx=\\lim_{n\\to\\infty}x_n$ of $(x_n)$ converges. We have thus constructed a \"limit\" to each bounded sequence, which respects linearity, and which coincides with the usual limit for converging sequences. In particular this proves that $(\\ell^\\infty)'\\supsetneq\\ell^1$. Indeed, if we had $L(x)=\\sum_nx_ny_n$ for some $(y_n)\\in\\ell^1$, then, if we define, for a fixed $m$ and $\\ell\\neq0$, $x_n:=0$ if $n<m$ and $x_n:=\\ell$ if $n\\geq m$, then we would get a contradiction : $$\\ell=L(x)=\\ell\\sum_{n\\geq m}y_n\\xrightarrow[m\\to\\infty]{}0.$$<\/p>\n

The space $S\\subset\\ell^\\infty$ above can be seen as the space of continuous functions on the Alexandrov compactification $\\overline{\\mathbb{N}}=\\mathbb{N}\\cup\\{\\infty\\}$ of $\\mathbb{N}$. The topology on this space is metrizable, and by a Riesz theorem, every linear form on $S$ can be seen as a measure on $\\overline{\\mathbb{N}}$. The linear form $L$ above corresponds then clearly to the Dirac mass at $\\infty$. This functional respects the additive structure but not necessarily the multiplicative one, in other words we do not have necessarily\u00a0$L(xy)=L(x)L(y)$, and similarly we do not have necessarily $L(f(x))=f(L(x))$ for all $f$.<\/p>\n

Representation.<\/strong><\/p>\n