there is a constant c such that for each distinct i and j. Exact frames A frame is an exact frame if no proper subset of the frame spans the inner product space. Each basis for an inner product space is an exact frame for the space (so a basis is a special case of a frame). Generalizations A Bessel Sequence is a set of vectors that satisfies only the upper bound of the frame condition. Continuous frame Suppose H is a Hilbert space, X a locally compact space, and is a locally finite Borel measure on X. Then a set of vectors in H, with a measure is said to be a Continuous Frame if there exists constants, such that for all . Example Given a discrete set and a measure where is the Dirac measure then the continuous frame property reduces to . and we see that continuous frames are indeed the natural generalization of the frames mentioned above. Just like in the discrete case we can define the analysis, synthesis, and frame operators when dealing with continuous frames. Continuous analysis operator Given a continuous frame the continuous analysis operator is the operator mapping to a sequence of coefficients . It is defined as follows: by . Continuous synthesis operator The adjoint operator of the continuous analysis operator is the continuous synthesis operator, which is the map by . Continuous frame operator The composition of the continuous analysis operator and the continuous synthesis operator is known as the continuous frame operator. For a continuous frame , it is defined as follows: by . Continuous dual frame Given a continuous frame , and another continuous frame , then is said to be a continuous dual frame of if it satisfies the following condition for all : . Dual frames The frame condition entails the existence of a set of dual frame vectors with the property that for any This implies that a frame together with its dual frame has the same property as a basis and its dual basis in terms of reconstructing a vector from scalar products. In order to construct a dual frame, we first need the linear mapping called the frame operator, defined as From this definition of and linearity in the first argument of the inner product, which, when substituted in the frame condition inequality, yields for each The frame operator is self-adjoint, positive definite,