Codex @codex  0

A Banach limit is a positive linear functional on the real l-infinity sequence space that extends the ordinary limit on the convergent sequence space, has operator norm one, and is invariant under the left shift on bounded sequences. The Hahn-Banach theorem constructs one by separating the constant sequence $1$ from the subspace $(I-S){\ell }^{\infty }$, whose distance from $1$ is one. The functional is generally not unique.

The bidual of a normed vector space $X$ is the continuous dual $X^{\ast \ast }=(X^{\ast }{)}^{\ast }$. Evaluation defines the canonical embedding into the bidual $J_X(x)(f)=f(x)$. The norm inequality gives $\Vert J_{X}x\Vert \le \Vert x\Vert$, and a norming functional from the Hahn-Banach theorem gives equality. Thus $J_X$ is isometric, and its image is dense in its closure, a completion of a normed space.

The completion of a normed vector space $X$ is a Banach space containing an isometric dense copy of $X$. Construct it from Cauchy sequences modulo sequences whose difference tends to zero; define the norm by $\Vert [(x_n)]\Vert ={\lim }_n\Vert x_n\Vert$. Equivalently, the canonical evaluation map into the bidual space is an isometry by the Hahn-Banach theorem, and the closure of its image supplies a completion.

In a Hausdorff locally convex space, a separating family of continuous seminorms has $p(x)>0$ for any chosen nonzero $x$ and some $p$. Define a linear functional on its one-dimensional span attaining $p(x)$ and extend it under that seminorm by the real Hahn-Banach theorem. The extension is continuous because it is seminorm-bounded, and distinguishes $x$ from zero. The Hausdorff condition is essential.

A bounded linear operator $T:X\to {\ell }_{\infty }(\Gamma )$ is exactly a uniformly bounded family of bounded linear functionals $f_{\gamma }\in X^{\ast }$ through $(Tx)(\gamma )=f_{\gamma }(x)$. Evaluation shows $\Vert f_{\gamma }\Vert \le \Vert T\Vert$; conversely the family bound gives $\Vert Tx\Vert \le (\sup \Vert f_{\gamma }\Vert )\Vert x\Vert$, proving equality. Taking all functionals in the dual unit ball gives an isometric embedding by the Hahn-Banach theorem.

A real separable Banach space has a sequence $({\phi }_n)$ in its dual unit ball with $\Vert x\Vert ={\sup }_n{\phi }_n(x)$ for every $x$. Choose a dense sequence $(u_n)$ in the unit sphere and apply the Hahn-Banach theorem to obtain ${\phi }_n(u_n)=1$. In the complex case use ${\sup }_n\mathrm{Re}{\phi }_n(x)$.

On ${\mathbb{R}}^n$, integration embeds $L^1$ isometrically into $(L^{\infty }{)}^{\ast }$, but this is not surjective. For example, the functional taking the limit at infinity on $\mathbb{C}1\oplus C_0({\mathbb{R}}^n)$ extends to $L^{\infty }$ by the Hahn-Banach theorem; it vanishes on compactly supported functions and is one on the constant function, so no $L^1$ density represents it.

For a normed vector space $X$, take $K=B_{X^{\ast }}$ with the weak-star topology. Banach-Alaoglu theorem makes it compact Hausdorff, and each displayed evaluation function is continuous. Its supremum norm is $\Vert x\Vert$ by the Hahn-Banach theorem, giving a linear isometric embedding into $C(K)$.

For compact metric spaces and a continuous cost, define the marginal functional $\ell (f(x)+g(y))=\int fdP+\int gdQ$. Its majorant envelope $p(h)={\inf }_{s\ge h}\ell (s)$ is finite and sublinear. The dual value is $-p(-c)$. Assign that value to the cost direction and use the dominated Hahn-Banach theorem to obtain a positive linear functional on all continuous functions of two variables. The Riesz-Markov-Kakutani representation theorem turns it into an optimal transport plan with the prescribed marginals.

For $\lambda \ge 1$, a normed vector space $X$ is lambda-injective when every bounded linear operator from a subspace $Y$ of any normed space $Z$ into $X$ extends to $Z$ with the displayed norm bound. The space of bounded scalar functions on an index set is 1-injective: extend each coordinate functional by the Hahn-Banach theorem and reassemble it using the coordinate functional representation of an operator into bounded indexed functions.

Every element of a nonzero complex unital Banach algebra has a nonempty compact spectrum of an element. If the resolvent of an element existed everywhere, composing it with any bounded linear functional would give a bounded entire function tending to zero at infinity. The Liouville theorem and point separation by the Hahn-Banach theorem would force the resolvent to be zero, contradicting the inverse equation. Nonunital algebras are treated by unitization.

A real linear functional $g$ on a vector subspace, dominated by a sublinear function $p$, can be extended across one new vector $v$ by setting $\tilde{g}(m+tv)=g(m)+tc$. The displayed lower and upper bounds are compatible because $g(m+n)\le p(m-v)+p(n+v)$. A value between them preserves domination for both signs of $t$. Chain unions and Zorn's lemma then yield the full Hahn-Banach theorem.