What does "matrix $A$ induces a linear operator on $\ell^2$" mean?

Question

In the description of Toeplitz matrices, it is said that:

A bi-infinite Toeplitz matrix $A$ induces a linear operator on $l^2$.

I do not know what does it mean to a matrix induce an operator. And also, what is space is $l^2$? Thanks in advance. I hope this is not too trivial.

Answer 1

Given your infinite matrix $\{a_{ij}\}_{i,j=1,2,\dots}$, given an element $x=(x_j)$ of $l_2$, the induced operator is defined by $$ A(x)_i=\sum\limits_{j=1}^\infty a_{ij}x_j $$