Fourier transform of a normalized vector
There are different conventions in scaling of the FFT, in MATLAB you need to scale it by $\sqrt{N}$, where $N$ is your number of samples. Saying it in matlabish:

clc, clear all
%% Create some sinusoidal signal with noise
t = linspace(0, 6*pi, 1000)+randn(1,1000);
s1 = sin(2*pi*t);
% Calculate the norm of s1
s1_norm = norm(s1);
display(sprintf('L2 norm of s1: %.2f', s1_norm))
%% Create signal s2 as the FFT of normalised s1
% Scale FFT by sqrt(N) - this is convention used in MATLAB
s2 = fft(s1)/sqrt(length(t));
% Calculate the norm of s2
s2_norm = norm(s2);
display(sprintf('L2 norm of s2: %.2f', s2_norm))