A Banach space is a complete normed vector space. Here, the norm is a function that measures the “length” of a vector $v$v, and it must satisfy certain properties like the triangle inequality. The distance between two vectors $u$u and $v$v is defined simply by the norm of their difference, $\|u – v\|$∥u−v∥. Importantly, Banach spaces don’t require the concept of an inner product — the norm alone defines the geometry.

A Hilbert space is a special kind of Banach space that comes equipped with an inner product, denoted as $\langle \cdot, \cdot \rangle$⟨⋅,⋅⟩. The norm in a Hilbert space is derived from this inner product, meaning the distance between two vectors $u$u and $v$v can be expressed as$$d(u, v) = \|u – v\| = \sqrt{\langle u – v, u – v \rangle}.$$d(u,v)=∥u−v∥=⟨u−v,u−v⟩​.

This inner product structure allows Hilbert spaces to naturally generalize Euclidean geometry into infinite dimensions. That’s why Hilbert spaces are foundational in areas like quantum mechanics and Fourier analysis — they give us a rigorous framework to handle concepts like angles, orthogonality, and projections in complex vector spaces.

What’s the application of Banach space?

Area of Mathematics	Typical Banach Space Used	Application Focus
Optimization/Numerical Methods	$L^1$ and $\ell^1$ Spaces	Used heavily in Compressed Sensing and LASSO Regression. The $L^1$ norm often promotes sparsity (solutions with many zero components), which is crucial for finding the simplest model that fits the data.
Approximation Theory	$C[a, b]$ Space (Continuous Functions)	Used in the Weierstrass Approximation Theorem. The focus is on uniform convergence (closeness measured by the maximum difference, the supremum norm), not convergence based on squared error.
Numerical Analysis/PDEs	$W^{k, p}$ Sobolev Spaces	These spaces define “weak derivatives” and are essential for proving the existence and uniqueness of solutions to Partial Differential Equations (PDEs), like the heat or wave equations, where $p \ne 2$ is often required.
Harmonic Analysis	$L^p$ Spaces ($p \ne 2$)	Used to study the convergence properties of Fourier series/integrals and other function expansions outside of the $L^2$ domain.