Adjoint of Inner Product Example

Let  
\[V\]
  be the vector space of polynomials over the field of complex numbers with inner product  
\[\]
   
\[f \cdot g= \int^1_0 f(t) \bar{g} (t) dt\]
  (where
\[\bar{g} (t)\]
) is the complex conjugate of  
\[g(t)\]
.
What is the adjoint of the linear operator  
\[M_f(g)=fg\]
  (multiplication by  
\[f\]
?)
If  
\[V\]
  is a linear product space and  
\[T\]
  a linear operator on  
\[V\]
  the adjoint  
\[T^*\]
  of  
\[T\]
  is a linear operator such that  
\[< T \alpha , \beta > = < \alpha , \beta T^* >\]
  for all  
\[\alpha , \; \beta \in V\]
.
We can write  
\[f(x)= \sum_k a_k x^k , \; \bar{f}(x)= \sum_k \bar{a}_k x^k\]
.
If  
\[f(x)= \bar{f}(x)\]
  then  
\[f\]
  is real.
The adjoint of 'multiplication by  
\[f\]
' is 'multiplication by  
\[\bar{f}\]
'.
\[\begin{equation} \begin{aligned} M_f (g) \cdot h &= \int^1_0 (fg)(t) \bar{h}(t)dt \\ &= \int^1_0 f(t)g(t) \bar{h}(t)dt \\ &= \int^1_0 g(t) \bar{ \bar{f} h}(t)dt \\ &= < g , \bar{f}h > \\ &= g \cdot M_{\bar{f} h} \end{aligned} \end{equation}\]