A quick counterexample to the proposed norm is the piecewise function
f(x) = 1 if x = 0.5
f(x) = 0 otherwise.
Here, f is not equal to the additive identity e(x) = 0, but the integral of the square of f is still equal to 0. A quick fix to this definition would be to restrict the L2 norm to onlly CONTINUOUS real-valued functions, instead of all real-valued functions.
A quick counterexample to the proposed norm is the piecewise function
f(x) = 1 if x = 0.5 f(x) = 0 otherwise.
Here, f is not equal to the additive identity e(x) = 0, but the integral of the square of f is still equal to 0. A quick fix to this definition would be to restrict the L2 norm to onlly CONTINUOUS real-valued functions, instead of all real-valued functions.