I am trying to understand dual spaces in real analysis. Let's look at the following:
Let $1\leq p<\infty$ and $$l_p:=\left\{ x=\{x_n\}\ |\ \sum_{n=0}^{\infty}|x_n|^p <\infty\right\}\quad \text{with norm}\quad \|x\|_p:=\left(\sum_{n=0}^{\infty}|x_n|^p\right)^{1/p}$$ and $$l_{\infty}:=\left\{ x=\{x_n\}\ |\ \sup_{n\in\mathbb N}|x_n| <\infty\right\}\quad \text{with norm}\quad \|x\|_{\infty}:=\sup_{n\in\mathbb N}|x_n|.$$ Then it follows that
$(l_p)^*\equiv l_q$ where $1/p+1/q=1$ with $q=\infty$ when $p=1$.
I think it's safe to assume that $^*$ represents the dual space and $\{x_n\}$ is a sequence.
As I understand it, the dual space of $l_p$ is the set of maps $(l_p,\|\cdot\|_p)\to(\mathbb R, |\cdot|)$. This set can be thought of as a set of relations between $l_p$ and $\mathbb R$, so each element in $(l_p)^*$ is a relation.
Now, $l_q$ where $1/p+1/q=1$ is a set $\left\{ \{x_n\}\ |\ \sum_{n=0}^{\infty}|x_n|^q <\infty\right\}$. I don't see how the elements of this set, which are infinite sequences, can be relations.
Can you please explain the conclusion?
Let $X = (x_n)_{n\in\Bbb{N}}$ be a sequence in $\ell^p$. Let $Y = (y_n)_{n\in\Bbb{N}}$ be a sequence in $\ell^q$. We reinterpret $(y_n)$ as a map in $(\ell^p)^*$ via $Y(X) = \sum_{n \in \Bbb{N}} x_n y_n$, yielding a map $Y:\ell^p \rightarrow\Bbb{R}$ or $\Bbb{C}$, as needed.