A Banach space is a complete normed vector space. It is a fundamental concept in functional analysis and has three defining properties: ## Vector Space A Banach space $X$ is a vector space over the field of real or complex numbers, meaning it is closed under addition and scalar multiplication. ## Normed Space There is a norm $||\cdot||: X \rightarrow [0, \infty)$ defined on $X$, which assigns a non-negative length (or size) to each vector in $X$. The norm has the following properties: - **Positive definiteness:** $||x|| = 0$ if and only if $x = 0$. - **Homogeneity:** $||\alpha x|| = |\alpha| \, ||x||$ for all scalars $\alpha$ and vectors $x$. - **Triangle inequality:** $||x + y|| \leq ||x|| + ||y||$ for all vectors $x$ and $y$. ## Completeness The space is complete with respect to the norm, which means that every Cauchy sequence in $X$ (a sequence where the distance between successive terms approaches zero) has a limit in $X$. In other words, if $\{x_n\}$ is a sequence in $X$ such that $||x_n - x_m|| \to 0$ as $n, m \to \infty$, then there exists an element $x \in X$ such that $x_n \to x$ in the norm $||\cdot||$. # Examples of Banach Spaces 1. **$L^p$ Spaces:** For $1 \leq p \leq \infty$, the $L^p$ spaces of measurable functions are Banach spaces with the $L^p$ norm: $ ||f||_{L^p} = \left( \int |f(x)|^p \, dx \right)^{\frac{1}{p}} $ When $p = \infty$, we use the essential supremum norm. 2. **$\ell^p$ Spaces:** For $1 \leq p \leq \infty$, the sequence spaces $\ell^p$ consist of all infinite sequences $(x_n)$ of real or complex numbers such that $\sum |x_n|^p < \infty$ (or bounded in the case of $\ell^\infty$). These spaces are also Banach spaces with the $\ell^p$ norm. 3. **$C([a, b])$:** The space of continuous functions on a closed interval $[a, b]$ with the sup norm, $ ||f||_{\infty} = \sup_{x \in [a, b]} |f(x)| $ is a Banach space. Banach spaces provide a broad framework for studying convergence, continuity, and boundedness in functional analysis.