at least one non-zero which would be orthogonal to all . If we insert into the frame condition, we obtain therefore , which is a violation of the initial assumptions on the lower frame bound. If a set of vectors spans V, this is not a sufficient condition for calling the set a frame. As an example, consider with the dot product, and the infinite set given by This set spans V but since , we cannot choose a finite upper frame bound B. Consequently, the set is not a frame. Applications In signal processing, each vector is interpreted as a signal. In this interpretation, a vector expressed as a linear combination of the frame vectors is a redundant signal. Using a frame, it is possible to create a simpler, more sparse representation of a signal as compared with a family of elementary signals (that is, representing a signal strictly with a set of linearly independent vectors may not always be the most compact form). Frames, therefore, provide robustness. Because they provide a way of producing the same vector within a space, signals can be encoded in various ways. This facilitates fault tolerance and resilience to a loss of signal. Finally, redundancy can be used to mitigate noise, which is relevant to the restoration, enhancement, and reconstruction of signals. In signal processing, it is common to assume the vector space is a Hilbert space. Special cases Tight frames A frame is a tight frame if A = B; in other words, the frame satisfies a generalized version of Parseval's identity. For example, the union of k disjoint orthonormal bases of a vector space is a tight frame with A = B = k. A tight frame is a Parseval frame (sometimes called a normalized frame) if A = B = 1. Each orthonormal basis is a Parseval frame, but the converse is not always true. A frame for is tight with frame bound A if and only if for all . Equal norm frame A frame is an equal norm frame (sometimes called a uniform frame or a normalized frame) if there is a constant c such that for each i. An equal norm frame is a unit norm frame if c = 1. A Parseval (or tight) unit norm frame is an orthonormal basis; such a frame satisfies Parseval's identity. Equiangular frames A frame is an equiangular frame if