I have a total of $3$ questions, my purpose is to clear up notation in the following question, since they are not introduced in the text and I cannot seem to find the notations online
Below, we denote:
$c_0$ is the real sequences converging to $0$
$\ell_1$ is the absolutely convergent real sequences
$X^*$ is the dual of the normed space $X$
Show that the bilinear form $<\cdot,\cdot>: c_0 \times \ell_1 \to \mathbb{R}$ given by $$ < \bar{x},\bar{y} > = \sum_{n=1}^{\infty} x_n y_n $$ is an Algebraic Duality. In other words, show that:
1. $<\cdot , \bar{y}> \in c_0^*$ for all $\bar{y} \in \ell_1$
2. $<\bar{x}, \cdot> \in \ell_1^*$ for all $\bar{x} \in c_0$
3. $<\cdot , \ell_1 >$ separates points of $c_0$
4. $<c_0, \cdot >$ separates points of $\ell_1$
5. Additionally, show that the norm of the bilinear form is $\leq 1$
So, my questions are:
1/2. What are the definitions of $<\cdot , \bar{y}>$ and $<\bar{x}, \cdot>$ (I dont have experience with bilinear forms)
3/4. What are the definitions of $<\cdot , \ell_1 >$ and $<c_0 , \cdot >$ (ie, putting a set instead of an element)
5. Usually the norm of a functional is $||f|| = \sup_{||x||\leq 1} |f(x)|$. But the bilinear form has two inputs, so how do I find norm? Is it $||<\cdot,\cdot>|| = \sup_{||x||,||y|| \leq 1} | <x,y> |$ ??
$\langle \cdot, \overline{y} \rangle$ denotes the functional $c_0 \rightarrow \mathbb{R}$ defined by $\overline{x} \mapsto \langle \overline{x}, \overline{y} \rangle$. Similarly $\langle \overline{x}, \cdot \rangle: \ell^1 \rightarrow \mathbb{R}$ via $\overline{y} \mapsto \langle \overline{x}, \overline{y} \rangle$.
$\langle \cdot, \ell^1 \rangle$ most likely means the set $\{ \langle \cdot, \overline{y} \rangle \vert \overline{y} \in \ell^1\}$. Similarly $\langle c_0, \cdot \rangle := \{ \langle \overline{x}, \cdot \rangle \vert \overline{x} \in c_0\}$
For the norm, the author probably refers to $\Vert \langle \cdot,\cdot \rangle \Vert := \sup \{\vert \langle \overline{x}, \overline{y} \rangle \vert : \overline{x} \in c_0, \overline{y} \in \ell^1, \Vert \overline{x} \Vert = \Vert \overline{y} \Vert = 1\}$