1.4 Examples and counterexamples of normed and Banach spaces

Banach spaces are complete normed vector spaces. They're crucial in functional analysis, providing a solid foundation for studying linear operators and continuous functions.

We'll look at examples of Banach spaces like bounded sequences and continuous functions on compact sets. We'll also explore counterexamples and compare Banach spaces to general normed spaces.

Examples of Banach spaces

• Space of bounded sequences $(l^{\infty}, \|\cdot\|_{\infty})$
  • Contains all sequences of real or complex numbers that are bounded
  • Norm is the supremum of the absolute values of the sequence elements: $\|x\|_{\infty} = \sup_{n \in \mathbb{N}} |x_n|$
  • Forms a Banach space as it is complete under this norm
  • Example: The sequence $(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots)$ belongs to $l^{\infty}$
• Space of continuous functions on a compact set $(C(K), \|\cdot\|_{\infty})$
  • $K$ represents a compact subset of a metric space
  • Includes all continuous functions $f: K \to \mathbb{R}$ or $\mathbb{C}$
  • Norm is the maximum absolute value of the function over its domain: $\|f\|_{\infty} = \sup_{x \in K} |f(x)|$
  • Forms a Banach space as it is complete under this norm
  • Example: The space of continuous functions on the closed interval $[0, 1]$ is a Banach space

Counterexamples to Banach spaces

• Space of continuous functions on an open interval $(C(a,b), \|\cdot\|_{\infty})$
  • Contains all continuous functions $f: (a,b) \to \mathbb{R}$ or $\mathbb{C}$
  • Norm is the maximum absolute value of the function over its domain: $\|f\|_{\infty} = \sup_{x \in (a,b)} |f(x)|$
  • Does not form a Banach space as it is not complete under this norm
  • Counterexample: The sequence of functions $f_n(x) = x^n$ on $(0,1)$ is Cauchy but does not converge to a continuous function on $(0,1)$
• Space of polynomials $(P, \|\cdot\|)$
  • Includes all polynomials with real or complex coefficients
  • Norm can be defined in various ways, such as $\|p\| = \sup_{x \in [0,1]} |p(x)|$
  • Does not form a Banach space as it is not complete under any norm
  • Counterexample: The sequence of polynomials $p_n(x) = \sum_{k=0}^n \frac{x^k}{k!}$ is Cauchy but converges to the exponential function $e^x$, which is not a polynomial

Completeness in specific spaces

• Space of polynomials $(P, \|\cdot\|)$
  • Not complete under any norm, as demonstrated by the previous counterexample
• Space of rational functions $(R, \|\cdot\|)$
  • Contains all rational functions of the form $f(x) = \frac{p(x)}{q(x)}$, where $p$ and $q$ are polynomials and $q \neq 0$
  • Norm can be defined in various ways, such as $\|f\| = \sup_{x \in [0,1]} |f(x)|$
  • Does not form a Banach space as it is not complete under any norm
  • Counterexample: The sequence of rational functions $f_n(x) = \frac{1}{1+nx^2}$ is Cauchy but converges to the function $f(x) = \begin{cases} 1, & x = 0 \\ 0, & x \neq 0 \end{cases}$, which is not a rational function

Comparison of normed vs Banach spaces

• Completeness
  • Banach spaces are always complete under their respective norms
  • Not all normed spaces are complete, as shown by the counterexamples
• Dimension
  • Banach spaces can be finite-dimensional ($\mathbb{R}^n$ with any norm) or infinite-dimensional ($l^{\infty}$ and $C(K)$)
  • Normed spaces can also be either finite-dimensional or infinite-dimensional
• Separability
  • Some Banach spaces are separable ($C([0,1])$ and $l^p$ for $1 \leq p < \infty$), while others are not ($l^{\infty}$ and $L^{\infty}([0,1])$)
  • Separability is not directly related to completeness, as both Banach and non-Banach spaces can be separable or non-separable
• Duality
  • The dual space of a normed space is always a Banach space
  • The dual of a Banach space is also a Banach space
  • The dual of a non-Banach space may not be complete under the operator norm