Inflation the scalar product (inner product) of the vector of price changes and the vector of basket quantities, with the former being uniform for the entire market, but the latter being different for different consumers, which is why an "average" basket is chosen for the official inflation number.
This is incorrect. You're confusing an estimator (the CPI) with the actual quantity being estimated (inflation). For example, the GDP deflator, which is another estimator of inflation, is not a weighted average.
