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Equation for a Real Valued Linear Function on Rn

Theorem
If  
\[\omega_{\mathbf{x}}\]
  is a real valued function on  
\[\mathbb{R}^n\]
  then  
\[\omega_{\mathbf{x}}(\mathbf{a}) = \sum_{i=1}^n f_i (\mathbf{x}_0 ) dx_i (\mathbf{a}) \]
  for all  
\[\mathbf{a} \in \mathbb{R}^n\]
  where  
\[f_i (\mathbf{x}_0) \]
  are real numbers.
Proof
\[\omega (\mathbf{x}) \]
  is a real valued linear function on  
\[\mathbb{R}^n\]

Any vector  
\[\mathbf{a} \in \mathbb{R}^n\]
  can be written  
\[\mathbf{a}=a_1 \mathbf{e}_1 + ... + a_n \mathbf{e}_n \]
.
The  
\[a_i\]
  are components of  
\[\mathbf{a}\]
  and the  
\[\mathbf{e}_i\]
  are the base vectors.
Since  
\[\omega_{\mathbf{x}}\]
  is a real valued function,
\[\begin{equation} \begin{aligned} \omega_{\mathbf{x}_0}(\mathbf{a}) &= \omega_{\mathbf{x}_0}(a_1 \mathbf{e}_1 + ... + a_n \mathbf{e}_n) \\ &=a_1 \omega_{\mathbf{x}_0} (\mathbf{e}_1) +...+ a_n\omega_{\mathbf{x}_0} (\mathbf{e}_n) \end{aligned} \end{equation}\]

For a fixed  
\[x_0\]
  all the  
\[ \omega_{\mathbf{x}_0} (\mathbf{e}_i)\]
  are real numbers hence  
\[\omega_{\mathbf{x}}(\mathbf{a}) = sum_{i=1}^n f_i (\mathbf{x}_0 ) dx+i (\mathbf{a}) \]
  for all  
\[\mathbf{a} \in \mathbb{R}^n\]
  then  
\[f_i (\mathbf{x}_0) \]

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