[In a CNN, does each new filter have different weights for each input channel, or are the same weights of each filter used across input channels?](https://ai.stackexchange.com/questions/5769/in-a-cnn-does-each-new-filter-have-different-weights-for-each-input-channel-or)

This question helps me a lot.

Let, I have RGB input image. (3 channels)
Then each filter has n×n weights for one channel.
It means, actually the filter has totally 3×n×n weights.

For channel R, it has own n×n filter.

For channel G, it has own n×n filter.

For channel B, it has own n×n filter.

After inner product, add them all to make one feature map.
Am I right?

And then, my question starts here.
For some purpose, I will only use greyscale images as input.
So the input images always have the same values for each RGB channel.

Then, can I reduce the number of weights in the filters?
Because in this case, using three different n×n filters and adding them is same with using one n×n filter that is the summation of three filters.

Does this logic hold on a trained network?
I have a trained network for RGB image input, but it is too heavy to run in real time.
But I only use the greyscale images as input, so it seems I can make the network less heavy (theoretically, almost 1/3 of original).

I'm quite new in this field, so detailed explanations will be really appreciated.
Thank you.