Important Vector Spaces
Contents
4.9. Important Vector Spaces#
In this section, we will list some important vector spaces which occur frequently in analysis and optimization.
4.9.1. The Vector Space of Real Valued Functions#
Definition 4.105 (The vector space of (total) real valued functions)
Let \(X\) be a non-empty set. Let \(\FFF (X, \RR)\) be the set of real valued total functions on \(X\). The set \(\FFF (X, \RR)\) is a vector space over the scalar field of \(\RR\) with the definitions following Definition 2.47:
Vector addition: If \(f,g \in \FFF (X, \RR)\), then \(h = f + g\) is defined as:
Scalar multiplication: if \(\alpha \in \RR\) and \(f \in \FFF (X, \RR)\), then \(h = \alpha f\) is defined as:
Additive identity: There exists a function \(\bzero \in \FFF (X, \RR)\) given by:
4.9.2. The Vector Space of Bounded Functions#
It was discussed earlier in Example 3.20.
Recall from Definition 2.50 that a real valued (total) function \(f: X \to \RR\) is called bounded if there exists a number \(M \geq 0\) (depending on \(f\)) such that
Definition 4.106 (The vector space of bounded functions)
Let \(X\) be a non-empty set. Let \(B(X)\) be the set of bounded functions on \(X\). The set \(B(X)\) is a vector space of bounded functions over the scalar field of \(\RR\) with the following operations:
Vector addition: If \(f,g \in B(X)\), then \(h = f + g\) is defined as:
Scalar multiplication: if \(\alpha \in \RR\), then \(h = \alpha f\) is defined as:
Definition 4.107 (Sup norm for the space of bounded functions)
The standard norm for \(B(X)\) is defined for any \(f \in B(X)\) as:
This norm is known as sup norm and often written as \(\| f \|_{\infty}\).
Definition 4.108 (Metric induced by the norm)
The standard metric induced by the standard norm for \(B(X)\) is defined for any \(f,g \in B(X)\) as:
Theorem 4.126 (\(B(X)\) is complete)
The normed vector space \(B(X)\) is complete. Thus, \(B(X)\) is a Banach space.
Proof. See Example 3.20 for the detailed proof.