Dot Product (Inner Product) of Vectors :

In simple language, the Dot product (Inner product or Scalar product) is a binary operation which takes two vectors and returns a scalar quantity. The Dot product of two vectors X and Y, both of 'N' dimensions is a scalar which does not depend on the choice of the orthogonal system with N directions. It is the Projection of one vector on the other i.e. component of one vector along another vector. By its very definition, dot product of a vector with itself is always non-negative and is the square of its magnitude.

Take 2 vectors, X = ( x[1], x[2], ... , x[N] )  &  Y = ( y[1], y[2], ... , y[N] ).  Here X and Y in general can be complex. Then the dot product of X with Y is given by:

What is the purpose of taking complex conjugates of components of Y?
Inner product of a vector with itself must be non-negative by definition as any vector is wholly contained in itself. If X is any complex vector this condition requires complement of Y to be taken in the above definition.

Conditions for a function to be an inner product in a vector space:
 An operation <X, Y> between two vectors X and Y can be called an inner product if and only if it satisfies the following conditions:

Now lets define the inner product for continuous and discrete time as shown below. Clearly, each of these definitions satisfies the necessary conditions for it to be described as an inner product.

Continuous Time: Consider X(t) and Y(t) as two signals in continuous time.

Provided the integral exists

Discrete Time: Consider X[n] and Y[n] as two signals in discrete time.

Compare this with the definition of dot product for two finite dimensional vectors