91Ó°ÊÓ

A Banach space is a type of vector space that is complete with respect to a norm. This means that every Cauchy sequence (a sequence where the elements become arbitrarily close to each other) in the space converges to a point within the space. Banach spaces are essential in functional analysis. They allow us to conduct detailed studies of vector spaces with a norm, especially infinite-dimensional ones. Examples of Banach spaces include spaces like \( l^p \) spaces (for \(1 \leq p \leq \infty \)) and the space of continuous functions on a closed interval, denoted by \( C([a, b]) \). One particular Banach space discussed in the exercise is \(c_0\), the space of all sequences converging to zero, normalized with the supremum norm.

The dual of a Banach space is another Banach space consisting of all continuous linear functionals on the space. A dual space ‘X*’ is separable if there exists a countable dense subset in \( X* \). This means we can approximate any element in the dual space arbitrarily well using only a countable set of functionals. Separable spaces often simplify the study of infinite-dimensional spaces by giving us a 'small' representative subset to work with. In our exercise, the Banach space \( c_0 \) is used, which has a separable dual \( \ell^1 \). This plays a critical role in determining whether the basis \( \{e_i\} \) is shrinking or non-shrinking.

A Schauder basis is a sequence of vectors such that any vector in the Banach space can be written uniquely as a convergent series of these basis vectors. However, not all Schauder bases have the same properties. A basis is called non-shrinking if the biorthogonal functionals corresponding to this basis do not form a Schauder basis for the dual space. In other words, if \( \{e_i^{*}\} \) is the sequence of biorthogonal functionals to the basis \( \{e_i\} \) in the dual space, then \( \{e_i\} \) is non-shrinking if and only if \( \{e_i^{*}\} \) does not form a Schauder basis for the dual space. In the exercise, the Schauder basis of \( \{e_i\} \) for the Banach space \( \c_{0} \) is demonstrated to be non-shrinking.

Biorthogonal functionals are used to define the relationship between a basis of a Banach space and its dual space. For a given Schauder basis \( \{e_i\} \) in a Banach space \(X\), the biorthogonal functionals \( \{e_i^{*}\} \) in the dual space \(X*\) are defined such that \( e_i^{*}(e_j) = \delta_{ij} \) where \( \delta_{ij} \) is the Kronecker delta (1 if \(i=j\) and 0 otherwise). These functionals provide a way to 'pick out' the coefficient of each basis element in any vector from the Banach space when expressed as a series of the basis vectors. In the context of \( c_0 \), the biorthogonal functional for a vector \( e_i\) is defined by \( e_i^{*}(x) = x_i \).

The supremum norm, often used in spaces of functions or sequences, measures the 'largest value' that elements of the sequence or function take. Specifically, for a sequence \( x = (x_n) \), the supremum norm is defined as: \[ \|x\| = \sup_{n} |x_n| \] This norm is particularly useful in the space \(c_0\), the space of all sequences that converge to zero. The supremum norm allows us to define a concept of size or length of a sequence in a way that is useful for analysis. It ensures the space is complete, making \(c_0\) a Banach space. When sequences converge to zero in \(c_0\), their supremum norm goes to zero as well.