Algebraic dual space of $c_{00}$

Question

Let $c_{00}$ be the space of real sequences that has finite non-zero elements. I do not know what is the algebraic dual (space of all linear forms on $c_{00}$) of $c_{00}$. Thank you for all construction.

Answer 1

For each $\varphi \in c_{00}'$, consider $$ y_j := \varphi(e_j) $$ where $e_j$ denotes the $j^{th}$ term in the standard basis of $c_{00}$. Then $y := (y_j)$ is merely a sequence of complex numbers with the property that $$ \varphi((x_n)) = \sum x_ny_n \quad\forall (x_n) \in c_{00} $$ Hence the algebraic dual would be the space of all sequences of complex numbers!