Function Spaces and Legal Operator Algebra

Conceptual map

III-10.01Spaces and norms
III-10.02Operator domains
III-10.03Meaning of an inverse
III-10.04Boundedness and theorem tools
III-10.05Composition and graph cores

1. An operator formula begins with spaces

A symbol such as $T^{-1}y$ has content only after the domain of $$T$$ , its range, and the norms on both spaces have been fixed.

Let $$X$$ and $$Y$$ be normed spaces and $T:D(T)\subseteq X\to Y$ a linear operator. A full-domain map $T:X\to Y$ is bounded when a finite constant $$M$$ satisfies $\lVert Tx\rVert_{Y}\le M\lVert x\rVert_{X}$ . If $$T$$ is injective, the inverse is the map

\[T^{-1}:\operatorname{range}(T)\to D(T).\](1)

Definition 1 · Legal composition

The composition $$ST$$ is defined on $\{x\in D(T):Tx\in D(S)\}$ . Algebraic cancellation involving $T^{-1}$ is valid only on vectors that belong to $\operatorname{range}(T)$ and to every subsequent operator domain.

2. Bounded inversion is a lower-bound property

Proposition 1 · Stability of an injective operator

Let $T:X\to Y$ be bounded and injective. Its inverse on $\operatorname{range}(T)$ is bounded exactly when a constant $$c>0$$ satisfies

\[\lVert Tx\rVert _{Y}\ge c\lVert x\rVert _{X} \quad \text{for every} x\in X.\](2)

If $$X$$ is Banach, condition (2) also implies that $\operatorname{range}(T)$ is closed in $$Y$$ .

Proof. If $\lVert T^{-1}\rVert \le M$ , then $\lVert x\rVert=\lVert T^{-1}Tx\rVert \le M\lVert Tx\rVert$ , giving (2) with $$c=1/M$$ . Conversely, (2) gives $\lVert T^{-1}y\rVert \le c^{-1}\lVert y\rVert$ on the range. If $Tx_{n}$ converges, (2) makes $x_{n}$ Cauchy; completeness supplies a limit whose image is the proposed range limit. ∎

3. An injective compact operator with unstable inverse

Take $X=Y=\ell ^{2}$ and define $Te_{k}=k^{-1}e_{k}$ . Then

\[(Tx)_{k}=x_{k}/k, \quad \operatorname{range}(T)=\{y\in \ell ^{2}: \sum k^{2}|y_{k}|^{2}<\infty \}.\](3)

The map is bounded, compact, and injective. Yet $\lVert T^{-1}e_{k}\rVert=k$ , so the inverse norm on successively larger coordinate bands diverges. The sequence $y_{k}=1/k$ belongs to $\ell ^{2}$ , while its formal inverse is the constant sequence and lies outside $\ell ^{2}$ . Thus $$y$$ is outside the range in (3).

In an $$N$$ -coordinate truncation, the inverse norm is exactly $$N$$ . Noise $\delta e_{N}$ in the measured object becomes $N\delta e_{N}$ after inversion. With $\delta=10^{-4}$ and $$N=10$$ ,000, a measurement perturbation of size $10^{-4}$ creates a latent error of size one.

4. Unbounded operators and graph cores

An unbounded operator can still be closed: convergence $x_{n}\to x$ and $Tx_{n}\to y$ then implies $x\in D(T)$ and $$Tx=y$$ . Its graph norm $\lVert x\rVert_{T}=\lVert x\rVert_{X}+\lVert Tx\rVert_{Y}$ records both the object and its image. A graph core is a subset dense in $$D(T)$$ under this norm.

Failure case · Ambient convergence used through an inverse

Convergence $y_{n}\to y$ in $$Y$$ gives no convergence of $T^{-1}y_{n}$ when the inverse is unbounded. The proof must supply convergence in the range norm or a regularization argument.

5. Implementation, exercises, and sources

For each displayed operator, create a type line containing domain, codomain, norm, range, and boundedness. For a numerical inverse, report the singular-value cutoff and the norm of the regularized inverse. A finite matrix result should be accompanied by its behavior as the discretization dimension grows.

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Exercises

Prove compactness of the diagonal operator in (3) by finite-rank truncation.
Characterize the range when the diagonal sequence is $2^{-k}$ and compute the inverse norm on the first $$N$$ coordinates.
Give operators $$S$$ and $$T$$ for which both are densely defined while $$D(ST)$$ is a strict subset of $$D(T)$$ .

Partial solutions

1. Let $T_{N}$ retain the first $$N$$ diagonal coordinates. It has finite rank and $\lVert T-T_{N}\rVert=1/(N+1)$ , which tends to zero; norm limits of finite-rank operators are compact. 2. The range is { $y:\sum 4^{k}|y_{k}|^{2}<\infty$ }, and the first- $$N$$ inverse norm is $2^{N}$ .

John B. Conway, A Course in Functional Analysis, Chapters II–III.Bounded operators, closed range, and compactness.
Walter Rudin, Functional Analysis, Chapters 2, 4.Banach-space operator theorems.
Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of Inverse Problems, Chapter 2.Ill-posed inversion and regularization.

6. Audit checkpoint

State every domain, codomain, norm, range, inverse domain, and composition domain. Report whether the inverse is bounded, closed, regularized, or available only in a finite truncation.

7. Scope boundary

The chapter establishes operator-domain discipline for Banach and Hilbert space inverse problems. Operator algebras and general spectral calculus are outside its boundary.