Just as the empty sum is 0 and the empty product is 1, the empty norm should be 0 in my opinion. In numpy, this convention has been implemented and is unit-tested for the default vector and matrix norms:

    def test_empty(self):
        assert_equal(norm([]), 0.0)
        assert_equal(norm(array([], dtype=self.dt)), 0.0)
        assert_equal(norm(atleast_2d(array([], dtype=self.dt))), 0.0)

Some of the non-default empty norms raise exceptions. The induced matrix 2-norm raises exceptions for two reasons -- first, attempting to compute the singular values of a size-0 matrix raises an exception, and second, even if the the empty array of singular values were to be returned, attempting to compute the max of an empty sequence also raises an exception. The empty induced matrix 1-norm and np.inf-norm each also raise an exception because they try to compute an empty max.