Normed and Banach Spaces

Functional analysis is linear algebra in infinite dimensions, where a vector space carries a norm (a notion of length) and one demands completeness (all Cauchy sequences converge). The resulting Banach spaces are the arena for operator theory, and their four "big theorems" are the structural backbone of the subject. This page opens the Functional Analysis folder and extends analysis/metric-spaces.md from finite ${\mathbb{R}}^n$ into function spaces.

Normed spaces

A norm on a vector space $X$ (over $\mathbb{R}$ or $\mathbb{C}$) is a map $\Vert \cdot \Vert :X\to [0,\infty )$ with $$\Vert x\Vert =0\iff x=0,\Vert \lambda x\Vert =|\lambda |\Vert x\Vert ,\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert (\text{triangle}).$$ A norm induces a metric $d(x,y)=\Vert x-y\Vert$, so all the topology of metric spaces applies. In finite dimensions all norms are equivalent and the space is automatically complete; in infinite dimensions neither holds, and completeness must be imposed.

Banach spaces

A Banach space is a normed space that is complete: every Cauchy sequence converges to a limit in the space. Completeness is what lets analysis proceed — limits, series, fixed points, and integrals of vector-valued functions all require it. Canonical examples:

• ${\ell }^p$ ($1\le p\le \infty$): sequences with $\sum |x_n{|}^p<\infty$ (or bounded, for $p=\infty$), norm $\Vert x{\Vert }_p=(\sum |x_n{|}^p{)}^{1/p}$.
• $L^p(\Omega )$: (equivalence classes of) measurable functions with $\int |f{|}^p<\infty$, norm $\Vert f{\Vert }_p=(\int |f{|}^p{)}^{1/p}$. Completeness is the Riesz–Fischer theorem. The case $p=2$ is a Hilbert space (hilbert-spaces.md).
• $C(K)$: continuous functions on a compact set with the sup norm.

These are genuinely infinite-dimensional and are the state and observable spaces of quantum theory.

Bounded linear operators

A linear map $T:X\to Y$ between normed spaces is bounded if $$\Vert T\Vert =\underset{x\ne 0}{\sup }\frac{\Vert Tx\Vert }{\Vert x\Vert }<\infty .$$ For linear maps bounded ⇔ continuous — a clean equivalence special to linearity. The bounded operators $\mathcal{B}(X,Y)$ themselves form a normed space, complete whenever $Y$ is. (In infinite dimensions unbounded linear operators exist and are unavoidable in physics — the momentum and energy operators — which is why unbounded-operators.md is a separate story.)

Dual spaces

The dual space $X^{\ast }=\mathcal{B}(X,\mathbb{F})$ is the space of bounded linear functionals on $X$ — the continuous "measurements" of vectors. Duals are central:

• $({\ell }^p{)}^{\ast }\cong {\ell }^q$ and $(L^p{)}^{\ast }\cong L^q$ with $\frac{1}{p}+\frac{1}{q}=1$;
• the bra $⟨\varphi |$ of Dirac notation is exactly a dual vector (a functional $|\psi ⟩\mapsto ⟨\varphi |\psi ⟩$);
• weak convergence ($x_n\to x$ tested against all functionals) is coarser than norm convergence and is the natural notion in infinite dimensions.

The four big theorems

Four foundational theorems (three resting on the Baire category theorem) organize the whole subject:

• Hahn–Banach. Bounded functionals extend from a subspace to the whole space without increasing norm — so $X^{\ast }$ is always rich enough to separate points.
• Uniform boundedness (Banach–Steinhaus). A family of operators bounded pointwise is bounded in norm.
• Open mapping theorem. A surjective bounded operator between Banach spaces is open; hence a bounded bijection has a bounded inverse (bounded inverse theorem).
• Closed graph theorem. A linear operator with closed graph is bounded — the practical test for boundedness, and the reason "closability" matters for unbounded operators.

References

• Reed & Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Ch. II–III.
• Rudin, Functional Analysis, Ch. 1–5.
• Kreyszig, Introductory Functional Analysis with Applications.