Normed vector space & Schauder basis exercise

I am taking a new course in functional analysis, so appreciate any help for this exercise. Let the set of sequences $l_p = \{a ∈ R^N: \sum^∞_{i=1} |a_i|^p< ∞\}$. I wish to determine or show:

(i) $l_p$ is a vector subspace of $R^N$. For this, I just have to show closed under scalar multiplication and addition?

(ii) $l_p$ has no countable basis? Do I need to consider some linearly independent vector to prove this?

(iii) The map $||a||_p := (\sum_i|a_i|^p )^{1/p}$ defines a norm on $l_p$. The main thing here is I can use Minkowski’s inequality? And to prove it, start with $|x + y|^p = |x + y| · |x + y|^{p−1} ≤ |x| · |x + y|^{p−1} + |y| · |x + y|^{p−1}$ followed by an application of Holder’s inequality?

(iv) The normed vector space $(l_p, || − ||_p)$ is not Banach? So I have to find a cauchy sequence with limit not in $l_p$ right?

(v) The “standard basis” elements $e_i$ form a topological spanning set of $l_p$ so that the algebraic linear span of the $e_i$’s is dense in $l_p$. In other words, I want to prove it is a Schauder basis right because the sum converge in $l_p$?

Solution 1:

$1).\ $ Suppose $a=\sum^∞_{i-1} |a_i|^p< ∞$ and $b=\sum^∞_{i-1} |b_i|^p< ∞.$ Then, the Minkowski inequality applies to show that, since $\|a\|_p=a^{1/p}<\infty;\ \|b\|_p=b^{1/p}<\infty$ so $\|a+b\|_p\le \|a\|_p+\|b\|_p<\infty$ which implies that $a+b\in \ell^p$. Now, if $\alpha\in \mathbb R$, then $\|\alpha a\|_p=\alpha^{1/p}\|a\|_p<\infty,$ so $\alpha a\in \ell^p.$

$2).\ $ It is enough to show that there is an uncountable linearly independent set of vectors in $\ell^p$. The trick I know is the following: Since $|\mathbb N|=|\mathbb Q|$, we can assume that $l^p = \{a ∈ R^{\mathbb Q}: \sum^∞_{i-1} |a_i|^p< ∞\},\ $ that is, functions from $\mathbb Q$ to $\mathbb R.$ Now define, for each $x\in \mathbb R,\ f_x:\mathbb Q\to \mathbb R$ by $f_x(q)=\chi_E(q)$, where $E=\{q\in \mathbb Q:q<x\}.$ Then, $\{f_x:x\in \mathbb R\}$ is the required uncountable, linearly independent set.

As an aside, if you know the Baire category theorem, you can show that no infinite dimensional Banach space, $X$, can have a countable Hamel basis, for if $\{e_1, e_2,\cdots \}$ is such a basis, then, letting $F_n=$span$\{e_1,\cdots,e_n\},$ we have $X=\bigcup F_n$, a countable union of closed sets with empty interior, and this is a contradiction of Baire's theorem.

$3).\ $ The Minkowski inequality shows that the triangle inequality holds. And if $\alpha\in \mathbb R,\ $ it is very easy to show that $\|\alpha a\|_p=|\alpha|\cdot \|a\|_p.$ Finally, if $\|a\|_p=0,\ $ then $\left ( \sum |a_i|^p \right )^{1/p}=0\Rightarrow a_i=0;\ i=1,2,\cdots $ so $a=0.$

$4).\ \ell^p$ is complete and so Banach. Here is a sketch: take a Cauchy sequence $(a_n)\subseteq \ell^p$ and show that each coordinate $(a^i_n)_i$ converges to a real number $a_i:\ i=1,2,\cdots$ (because the coordinates themselves are Cauchy in $\mathbb R$ and $\mathbb R$ is complete.) Now, consider $a=(a_1,a_2,\cdots)$ and show that $a\in \ell^p$ and that $\|a-a_n\|\to 0$ as $n\to \infty.$

$5).\ $ You need to show that span$\{e_i:i\in \mathbb N\}$ is dense in $\ell^p$. So, let $a=(a_1,a_2,\cdots )\in \ell^p$ and consider $a_n=(a_1,a_2,\cdots, a_n,0,0,\cdots)=\sum^n_{i=1} a_ie_i.$ Then, $a-a_n=(0,0,\cdots, n+1,n+2,\cdots)=\sum^{\infty}_{i=n+1}a_i\Rightarrow \|a-a_n\|^p\le \sum^{\infty}_{i=n+1}|a_i|^p\to 0$ because $a\in \ell^p.$

Solution 2:

(i) And that it is non-empty, but that's trivial.

(ii) Assume that it had a countable basis and try to construct an element that cannot be written as a finite linear combination.

(iii) Yes, your attempt looks good.

(iv) It is Banach (assuming $p \geq 1$). You need to show completeness.

(v) Yes, exactly. Cannot be a Hamel basis because of (ii).