Banach Spaces

Definition: A Banach Space is a complete normed linear space.

Recall that a metric space $(X, d)$ is said to be complete if every Cauchy sequence in $X$ converges to a point in $x$. If $X$ is a normed linear space with norm $\| \cdot \|$ then we can define a metric on $X$ called the metric induced by the norm for all $x, y \in X$ by $d(x, y) = \| x - y \|$, and when we say a normed linear space is complete, we mean that it is complete with respect to the metric induced by the norm.

The simplest example of a Banach space is $\mathbb{R}$ with the Euclidean norm of the absolute value. It is easy to show that every sequence of real numbers converges to a real number in $\mathbb{R}$.

An example of a space that is not a Banach space is subspace that is the interval $(0, 1]$ with the Euclidean norm of the absolute value. The sequence $\displaystyle{ \left (\frac{1}{n} \right )_{n=1}^{\infty}}$ is a Cauchy sequence in $(0, 1]$ that converges to $0 \not \in (0, 1]$, and so $(0, 1]$ is not complete.

One of the largest classes of Banach spaces that we have already looked at are the Lebesgue spaces. Recall that for $1 \leq p \leq \infty$ and for $(X, \mathfrak T, \mu)$ a measure space, that $L^p(X, \mathfrak T, \mu)$ is a normed linear space. And by The Riesz-Fischer Theorem we have that every Cauchy sequence of functions in $L^p(X, \mathfrak T, \mu)$ converges in $L^p(X, \mathfrak T,\mu)$, and so $L^p(X, \mathfrak T, \mu)$ is a Banach space.