Frame of A Vector Space - Relation To Bases

Relation To Bases

If the set is a frame of V, it spans V. Otherwise there would exist at least one non-zero which would be orthogonal to all . If we insert into the frame condition, we obtain

A \| \mathbf{v} \|^{2} \leq 0 \leq B \| \mathbf{v} \|^{2} ;

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 and the infinite set given by

This set spans V but since we cannot choose . Consequently, the set is not a frame.

Read more about this topic:  Frame Of A Vector Space

Famous quotes containing the words relation to, relation and/or bases:

    “The proper study of mankind is man in his relation to his deity.”
    —D.H. (David Herbert)

    “You know there are no secrets in America. It’s quite different in England, where people think of a secret as a shared relation between two people.”
    —W.H. (Wystan Hugh)

    “The bases for historical knowledge are not empirical facts but written texts, even if these texts masquerade in the guise of wars or revolutions.”
    —Paul Deman (1919–1983)