Lagmental Vicfred

Annihilators Reverse Inclusion by Vicfred

tags: mathematics, vector spaces, linear maps, dual spaces

The annihilator of a subspace consists of all linear functionals vanishing on it. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

The data

A linear map \(T:V\to W\) is organized by its kernel \(\ker T\) and image \(\operatorname{im}T\). Quotients and duals express the same information without choosing bases.

$$ U^\circ=\{\lambda\in V^\ast:\lambda(u)=0\ \forall u\in U\} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ U_1\subseteq U_2\Longrightarrow U_2^\circ\subseteq U_1^\circ $$

Derivation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \dim U+\dim U^\circ=\dim V,\qquad(U^\circ)^\circ=U\quad\text{inside }V^{\ast\ast}\cong V $$

Invariant content

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad U^\circ=\{\lambda\in V^\ast:\lambda(u)=0\ \forall u\in U\},\\[5pt] \mathsf{C}\;&:\quad U_1\subseteq U_2\Longrightarrow U_2^\circ\subseteq U_1^\circ. \end{aligned} $$

Scope

Dimension formulas below assume finite-dimensional spaces. Infinite-dimensional vector spaces require cardinal arithmetic and may not identify naturally with their double duals.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] U_1\subseteq U_2\Longrightarrow U_2^\circ\subseteq U_1^\circ \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Wed 13 November 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.