Frames and Bases

Roy, Raju and D, Venku Naidu (2017) Frames and Bases. Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

In the study of vector spaces, one of the most important concepts is that of a basis. By helping of basis we know all about element of that vector spaces, also we know that every element of vector spaces can be obtained by uniquely linear combination of basis elements. However, the conditions to a basis are very restrictive: we require that the elements are linearly independent, and very often we even want them to be orthogonal with respect to inner product. This makes it hard or even impossible to find bases satisfying extra conditions, and this is the reason that one might wish to look for a more flexible tool. As an example, if we take R 3 ( R ) then we can find basis easily. But if we take l ∞ , c [ a,b ] then we can not find easily but it has basis. From this difficulty, we wish to look for more flexible tool. Frames are such type of tool. A frames for a vector space equipped with an inner product also allow each vector in the space to be written as a linear combination of the elements in the frame. But here linearly independent condition is not required.

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