Bases

Abstract

Bases play a prominent role in the analysis of vector spaces, as well in the finite-dimensional as in the infinite-dimensional case. The idea is the same in both cases, namely, to consider a family of elements such that all vectors in the considered space can be expressed in a unique way as superpositions of these elements. In the infinite-dimensional case, the situation is complicated: we are forced to work with infinite series, and different concepts of a basis are possible, depending on how we want the series to converge. For example, are we asking for the series to converge with respect to a fixed order of the elements (conditional convergence) or do we want it to converge regardless of how the elements are ordered (unconditional convergence)? We define the relevant types of bases in general Banach spaces in Section 3.1; the case of a basis in a Hilbert space is considered in Section 3.3. Sequences satisfying the Bessel inequality are considered in Section 3.2 and characterized in terms of an associated operator, the synthesis operator. In Section 3.4 we discuss the most important properties of orthonormal bases in Hilbert spaces; we expect the reader to have some basic knowledge about this subject. Section 3.5 deals with the Gram matrix and its relationship with Bessel sequences. In Section 3.6, one of the key subjects of the current book, namely, Riesz bases, is introduced and treated in detail; a subspace version of these is discussed in Section 3.7. Several characterizations of Riesz bases and Riesz sequences are provided. Orthonormal bases and Riesz bases both satisfy the Bessel inequality, which is the key to the observation that they deliver unconditionally convergent expansions and can be ordered in an arbitrary way.