Transpose of a linear map

Suppose now that ${\displaystyle u:X\to Y}$ is a weakly continuous linear operator between topological vector spaces ${\displaystyle X}$ and ${\displaystyle Y}$ with continuous dual spaces ${\displaystyle X^{\prime }}$ and ${\displaystyle Y^{\prime },}$ respectively. Let ${\displaystyle \langle \cdot ,\cdot \rangle :X\times X^{\prime }\to \mathbb {C} }$ denote the canonical dual system, defined by ${\displaystyle \left\langle x,x^{\prime }\right\rangle =x^{\prime }x}$ where ${\displaystyle x}$ and ${\displaystyle x^{\prime }}$ are said to be orthogonal if ${\displaystyle \left\langle x,x^{\prime }\right\rangle =x^{\prime }x=0.}$ For any subsets ${\displaystyle A\subseteq X}$ and ${\displaystyle S^{\prime }\subseteq X^{\prime },}$ let

$${\displaystyle A^{\circ }=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|x^{\prime }(a)\right|\leq 1\right\}\qquad {\text{ and }}\qquad S^{\circ }=\left\{x\in X:\sup _{s^{\prime }\in S^{\prime }}\left|s^{\prime }(x)\right|\leq 1\right\}}$$

denote the (absolute) polar of ${\displaystyle A}$ in ${\displaystyle X^{\prime }}$ (resp. of ${\displaystyle S^{\prime }}$ in ${\displaystyle X}$).

• If ${\displaystyle A\subseteq X}$ and ${\displaystyle B\subseteq Y}$ are convex, weakly closed sets containing the origin then ${\displaystyle {}^{t}u\left(B^{\circ }\right)\subseteq A^{\circ }}$ implies ${\displaystyle u(A)\subseteq B.}$[7]
• If ${\displaystyle A\subseteq X}$ and ${\displaystyle B\subseteq Y}$ then[4]

$${\displaystyle [u(A)]^{\circ }=\left({}^{t}u\right)^{-1}\left(A^{\circ }\right)}$$

and

$${\displaystyle u(A)\subseteq B\quad {\text{ implies }}\quad {}^{t}u\left(B^{\circ }\right)\subseteq A^{\circ }.}$$

• If ${\displaystyle X}$ and ${\displaystyle Y}$ are locally convex then[5]

$${\displaystyle \operatorname {ker} {}^{t}u=\left(\operatorname {Im} u\right)^{\circ }.}$$

Suppose ${\displaystyle X}$ and ${\displaystyle Y}$ are topological vector spaces and ${\displaystyle u:X\to Y}$ is a weakly continuous linear operator (so ${\displaystyle \left({}^{t}u\right)\left(Y^{\prime }\right)\subseteq X^{\prime }}$). Given subsets ${\displaystyle M\subseteq X}$ and ${\displaystyle N\subseteq X^{\prime },}$ define their annihilators (with respect to the canonical dual system) by[6]

${\displaystyle {\begin{alignedat}{4}M^{\bot }:&=\left\{x^{\prime }\in X^{\prime }:\left\langle m,x^{\prime }\right\rangle =0{\text{ for all }}m\in M\right\}\\&=\left\{x^{\prime }\in X^{\prime }:x^{\prime }(M)=\{0\}\right\}\qquad {\text{ where }}x^{\prime }(M):=\left\{x^{\prime }(m):m\in M\right\}\end{alignedat}}}$

and

${\displaystyle {\begin{alignedat}{4}{}^{\bot }N:&=\left\{x\in X:\left\langle x,n^{\prime }\right\rangle =0{\text{ for all }}n^{\prime }\in N\right\}\\&=\left\{x\in X:N(x)=\{0\}\right\}\qquad {\text{ where }}N(x):=\left\{n^{\prime }(x):n^{\prime }\in N\right\}\\\end{alignedat}}}$

• The kernel of ${\displaystyle {}^{t}u}$ is the subspace of ${\displaystyle Y^{\prime }}$ orthogonal to the image of ${\displaystyle u}$:[7]

$${\displaystyle \ker {}^{t}u=(\operatorname {Im} u)^{\bot }}$$

• The linear map ${\displaystyle u}$ is injective if and only if its image is a weakly dense subset of ${\displaystyle Y}$ (that is, the image of ${\displaystyle u}$ is dense in ${\displaystyle Y}$ when ${\displaystyle Y}$ is given the weak topology induced by ${\displaystyle \operatorname {ker} {}^{t}u}$).[7]
• The transpose ${\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }}$ is continuous when both ${\displaystyle X^{\prime }}$ and ${\displaystyle Y^{\prime }}$ are endowed with the weak-* topology (resp. both endowed with the strong dual topology, both endowed with the topology of uniform convergence on compact convex subsets, both endowed with the topology of uniform convergence on compact subsets).[8]
• (Surjection of Fréchet spaces): If ${\displaystyle X}$ and ${\displaystyle Y}$ are Fréchet spaces then the continuous linear operator ${\displaystyle u:X\to Y}$ is surjective if and only if (1) the transpose ${\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }}$ is injective, and (2) the image of the transpose of ${\displaystyle u}$ is a weakly closed (i.e. weak-* closed) subset of ${\displaystyle X^{\prime }.}$[9]

Let ${\displaystyle M}$ be a closed vector subspace of a Hausdorff locally convex space ${\displaystyle X}$ and denote the canonical quotient map by

$${\displaystyle \pi :X\to X/M\quad {\text{ where }}\quad \pi (x):=x+M.}$$

Assume ${\displaystyle X/M}$ is endowed with the quotient topology induced by the quotient map ${\displaystyle \pi :X\to X/M.}$ Then the transpose of the quotient map is valued in ${\displaystyle M^{\bot }}$ and

$${\displaystyle {}^{t}\pi$$ :(X/M)^{\prime }\to M^{\bot }\subseteq X^{\prime }}

is a TVS-isomorphism onto ${\displaystyle M^{\bot }.}$ If ${\displaystyle X}$ is a Banach space then ${\displaystyle {}^{t}\pi$ :(X/M)^{\prime }\to M^{\bot }} is also an isometry.[6] Using this transpose, every continuous linear functional on the quotient space ${\displaystyle X/M}$ is canonically identified with a continuous linear functional in the annihilator ${\displaystyle M^{\bot }}$ of ${\displaystyle M.}$

Let ${\displaystyle M}$ be a closed vector subspace of a Hausdorff locally convex space ${\displaystyle X.}$ If ${\displaystyle m^{\prime }\in M^{\prime }}$ and if ${\displaystyle x^{\prime }\in X^{\prime }}$ is a continuous linear extension of ${\displaystyle m^{\prime }}$ to ${\displaystyle X}$ then the assignment ${\displaystyle m^{\prime }\mapsto x^{\prime }+M^{\bot }}$ induces a vector space isomorphism

$${\displaystyle M^{\prime }\to X^{\prime }/\left(M^{\bot }\right),}$$

which is an isometry if ${\displaystyle X}$ is a Banach space.[6]

Denote the inclusion map by

$${\displaystyle \operatorname {In} :M\to X\quad {\text{ where }}\quad \operatorname {In} (m):=m\quad {\text{ for all }}m\in M.}$$

The transpose of the inclusion map is

$${\displaystyle {}^{t}\operatorname {In} :X^{\prime }\to M^{\prime }}$$

whose kernel is the annihilator ${\displaystyle M^{\bot }=\left\{x^{\prime }\in X^{\prime }:\left\langle m,x^{\prime }\right\rangle =0{\text{ for all }}m\in M\right\}}$ and which is surjective by the Hahn–Banach theorem. This map induces an isomorphism of vector spaces

$${\displaystyle X^{\prime }/\left(M^{\bot }\right)\to M^{\prime }.}$$