Inner product

The set of periodic signals with period T form a vector space.

We define the following inner product:

And the norm or magnitude of the signal is defined as:-

Now we consider the set of vectors , that belong to this vector space.                 (note )

We shall first show these vectors are mutually orthogonal. In other words we show that:-

Further, you may verify  : 

Thus, we have shown that this set of complex exponentials forms an orthogonal set in the vector space of all periodic signals with period T. Indeed, if we restrict ourselves to a certain class of signals in this vector space (those that satisfy the Dirichlet Conditions, which will be discussed in the next lecture), one can show that the above set of complex exponentials forms a basis for this class. i.e.: signals in this class can be expressed as a linear combination of these complex exponentials. In other words, such signals permit a Fourier Series representation.