So how are vector spaces going to help us in dealing with discrete time signals. Well, by now, it should be self-evident that finite-length signals live in CN. CN is the space of length and topples of complex numbers in vector notation, therefore we will indicate a finite length signal by the column vector, x0, x1 up to xN-1, the transpose operator here indicates that this is a column vector. So it's actually like so. All operations are well defined intuitive. We have addition and scalar multiplications defined for CN. The space of N-periodic signals is also equal to CN, because the amount of information of the periodic signal is equivalent to the amount of information of the finite-length signal of length N. Sometimes to stress explicitly the periodicity of the signal, we will indicate the vector space of periodic sequence with the notation tilde C of N, like so. In CN, the inner product is defined like this. It's the sum for small n that goes from zero to big N-1 of the conjugate of the elements of the first topple, times the corresponding element of the second tuple. This is well defined for all finite-length vectors, so all vectors that live in CN, therefore, all finite-length and periodic signals. What about infinite-length signals? We could extend the concept of CN to C infinity, but we would run into a problem. The inner product defined as now an infinite summation for an index that goes from minus infinity to plus infinity, may not explode even for simple signals as x of n equal to the unit's depth. If you take the self-inner product of this, you would get an infinite number of ones here in the summation, and this would be equal to infinity. So in order to make sure that this doesn't happen, in order to make sure that the inner product is always well defined, we will define a vector space for infinite-length sequences by requiring that all sequences in this vector space are square-summable. Namely, the sum of the squares of the elements in the sequence is less than infinity. If you remember the definition of energy of a signal, this is equivalent to requiring that all sequences that live in this vector space have finite energy. This is the space of square-summable sequences. And we indicate this with the notation l2 of Z. In l2 of Z, everything is well. We can still use the same vector notation that we used for finite-length signals. So we now have an infinite-column vector. However, many interesting signals are not in l2 of Z. The unit's depth, the constant, sines of the functions and so on, and so forth. We will learn how to deal with these pesky sequences without losing the vector formulism in due course. Finally, a last technicality. We have defined vector spaces in such a way that the standard operations, scalar multiplications and addition do not take us outside of the vector space. In another words, any linear combination of vectors is a member of the vector space. Now, if we have an infinite sequence of vectors in the vector space that converges to a limit, we want this limit to be in the vector space as well. If a vector space is closed under the limiting operation, we say that the vector space is complete. Now, this is a technical requirement that is never really used explicitly in practical signal processing, but will be useful to prove some fundamental results such as the sampling theorem. It's hard to come up with a simple example of a non-complete vector space, but to give you an idea of what completeness entail, you can consider for instance, the set of all rational numbers, Q. We can build converging sequences of rational numbers that do not converge to a rational number themselves. Famously, the series for the exponential function which is written here, converges to the number E, in spite of the fact that each number of the series is a rational number. When a vector space equipped with an inner product is also complete, we call the vector space a Hilbert space.
