Is my transform the essence of DFT? I'm someone just learning DSP, and want understand its essence. My transform is the simplest possible. Input signal is just one frequency: $256\textrm{ Hz}$. Sampling frequency is $2560\textrm{ samples/sec}$, so $10\textrm{ samples}$ correspond to one cycle (period). I correlate it with two testing frequencies: one equal to input and other double of it: $2\cdot 256\textrm{ Hz} = 512\textrm{ Hz}$. I correlate only $10\textrm{ samples}$ so correlation is not moving. 


1st correlation: Input samples are in first column and testing frequency's (equal) in second: 
 0              x  0              = 0
 0.587785252292 x  0.587785252292 = 0.345491502812
 0.951056516295 x  0.951056516295 = 0.904508497187
 0.951056516295 x  0.951056516295 = 0.904508497187
 0.587785252292 x  0.587785252292 = 0.345491502812
 0              x  0              = 0
-0.587785252292 x -0.587785252292 = 0.345491502812
-0.951056516295 x -0.951056516295 = 0.904508497187
-0.951056516295 x -0.951056516295 = 0.904508497187
-0.587785252292 x -0.587785252292 = 0.345491502812
        Sum of products (squares) = 5

Because square of every real number is positive, sum is big number. Divided by $10$ yields $0.5$. Given that it is mean of products of sines is significant number, so I conclude frequency $256\textrm{ Hz}$ does exist in the signal. 
2nd correlation. Input samples are in first column and testing frequency's in second: 
 0              x  0              =  0
 0.587785252292 x  0.951056516295 =  0.559016994374
 0.951056516295 x  0.587785252292 =  0.559016994374
 0.951056516295 x -0.587785252292 = -0.559016994374
 0.587785252292 x -0.951056516295 = -0.559016994374
 0              x  0              =  0
-0.587785252292 x  0.951056516295 = -0.559016994374
-0.951056516295 x  0.587785252292 = -0.559016994374
-0.951056516295 x -0.587785252292 =  0.559016994374
-0.587785252292 x -0.951056516295 =  0.559016994374
                  Sum of products =  0

Because half products are opposed to other half, sum is $0$. It means frequency $512\textrm{ Hz}$ does not exist in the signal. 


My question: is that the essence of DFT?
I think you could say that in simple terms this is the essence of the DFT. Just note that the DFT correlates against complex exponentials:

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{1}$$

You've just computed the (negative) imaginary part of $(1)$ (for $k=2$):

$$-X_I[k]=\sum_{n=0}^{N-1}x[n]\sin(2\pi nk/N)\tag{2}$$

For your example that's also sufficient because the real part is zero since your signal $x[n]$ is one full period of a sine wave. However, imagine you added a phase to your signal, then you would also need the real part of $(1)$. In the worst case you could choose a cosine for $x[n]$, which would give you a correlation of zero, even for the same frequency, if you just correlated with sine functions as in $(2)$.