Hi, I would like to apply edgeR TMM normalization without converting data format into "DGEList", in other words I would like to manipulate the raw count table. (By doing so I can get a feeling about the whole RNAseq analysis procedure).

May I ask that if I just do the following:

(Raw count table)*(norm.factor), where 
norm.factor=edgeR::calcNormFactors((Raw count table),method='TMM').

And then conduct downstream analysis. Does this procedure make sense? After downstream analysis, can I say that I have applied TMM method for normalisation?

If the above procedure does not make sense, how about the following procedure:

(Raw count table)/((norm.factor)*colSums((Raw count table))), where

colSums((Raw count table)) stands for library size (each column represents a sample or a cell)

Thank you very much!

Your second approach is closer to what edgeR does internally. Normalization is performed by conceptually dividing the counts by the effective library size. The effective library size is, in turn, defined as the product of the normalization factor and the library size for each sample, to account for composition biases.

In practice, normalization is achieved by scaling up the mean by the effective library size, rather than scaling down the observed count. This avoids distorting the mean-variance relationship from modifying the counts directly. I would not use the "normalized counts" for the DE analysis, only for stuff like clustering.

After downstream analysis, can I say that I have applied TMM method for normalisation?

Well, it depends on what downstream analysis you do, which you haven't told us anything about. How can we comment on things you don't tell us?