I have a hard time interpreting the adjoint of operators defined over Banach spaces

Let $T$ be an operator defined by $$T:X \to Y$$ where X and Y are Banach spaces. The adjoint of $T$, $T^*$ is defined by
$${T^*}:{Y^*} \to {X^*}$$ where ${X^*}$, ${Y^*}$ are the dual spaces. In other words, the adjoint maps linear functionals of $Y$ (elements of $Y^*$) into linear functionals of $X$ (elements of $X^*$). This is a bit hard to wrap one's head around it. Could someone give an example of an operator and its adjoint and how it maps linear functional into other linear functionals, or elaborate on the concepts.

Thanks in advance.

If $f\in Y^*$ then $f$ is a map from $Y$ to $\Bbb R$ or to $\Bbb C$. We define the map $g=T^*(f)$ from $X$ to $\Bbb R$ or to $\Bbb C$ by letting $g(x)=f(T(x))$ for each $x\in X.$ That is, $T^*(f)(x)=f(T(x)).$

Note that this is not $T^*(f(x)),$ which is meaningless. $T^*(f)$ is to be treated as a single symbol for one member of $X^*$.

For example if $X=Y=\Bbb R^2$ and $T(u,v)=(2v,u)$ for all $(u,v)$ then for any $f\in Y^*$ we have $T^*(f)(u,v)=f(T(u,v))=f(2v,u).$