I am relatively new to the field of computer vision and I have just learnt about the sobel operator. The sobel operator in the x direction is a convolution of the finite difference kernel $[1,0,-1]$ and the gaussian smoothing kernel $[1,2,1]$. Why is it the case that the smoothing kernel does not need to be normalised ?
For example, the vector below convolved with the image will result in pixel intensities that are higher than the original values. Eg, $[50,100,50]$ will result in the middle pixel getting a value of 300 which is not the intended effect of smoothing. If normalisation is applied, then the middle pixel would get a value 75, which blurs the image. \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}
I hope my question was clear in the sense that i don't see how applying $[1,2,1]$ filter results in blurring without normalisation.