Bounded linear operator

Mathematical statement

A bounded linear operator T:V→W admits C≥0 such that every x satisfies norm_W(Tx)≤C*norm_V(x). This slice uses exact induced-operator-norm certificates for integer matrices under l1 or linfinity norms.

Representative formula (exact TeX)

O(T)

Assumptions

• Source and target are explicit finite-dimensional real coordinate spaces with fixed norm names at both ends.
• The l1 formula uses the maximum absolute column sum and linfinity the maximum absolute row sum; an instance pass does not replace a universal certificate.

Symbol scope

• O — bounded-operator record
• T — linear map
• C — uniform bound constant

Intuition, definition, and misconceptions

The same C must control every input.

A nonoptimal C can certify boundedness without being the operator norm.

For a matrix in finite-dimensional named norms, the induced operator norm can be calculated exactly.

Finitely many inputs, an apparently bounded range, or any usable upper bound do not supply the definition of the exact operator norm.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Linear map between real vector spaces — The map must first qualify as linear.
• Real normed vector space — Source and target norms must be explicit.
• Usual order on the real numbers — The uniform norm inequality uses a nonnegative ordering.

Verification boundary

Only finite-dimensional integer matrices and named norms are checked; no arbitrary infinite-dimensional operator or spectral conclusion follows.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add continuous linear operators, duality and spectral theory.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.