If statement running but not processing

I've been scratching my head all morning trying to understand why the following doesn't seem to work. The idea here is that if there is no column provided, run 1 set of code and if there is, then run another.

Let's say the value of df just holds 1 value = 'A'

def function_name(df, col):
    if col == None:
        df = df.str.lower()

    else:
        df[col] = df[col].str.lower()

function_name(df, None)

Expected Results: 'a' Current Results: 'A'

If I was to run function_name(df, 'A'):

Expected Results: 'a' Current Results: 'a'

Ideally when running the function, since None was passed in it should return whatever commands I passed through but currently, it's acting as if nothing is happening. When I debug by printing it, I can see that the code is doing the 'stuff' but the function itself isn't resulting in whatever commands was run. Any thoughts?