Bounded Linear Operators | Mathematics

A linear operator $T : X \to Y$ between normed spaces is bounded if there exists $C \geq 0$ such that $\|Tx\|_Y \leq C\|x\|_X$ for all $x \in X$. The operator norm is

$\|T\| = \sup\{\|Tx\|_Y : \|x\|_X \leq 1\} = \sup\{\|Tx\|_Y : \|x\|_X = 1\}$

Proposition 3.1. A linear operator is bounded if and only if it is continuous.

Proposition 3.2. $T$ is bounded if and only if it maps bounded sets to bounded sets.

The space $\mathcal{B}(X, Y)$ of all bounded linear operators from $X$ to $Y$ is a Banach space when $Y$ is complete, with the operator norm.

Example 1. The identity operator $I : X \to X$ has $\|I\| = 1$.

Example 2. The zero operator $0 : X \to Y$ has $\|0\| = 0$.

Example 3. Let $T : \ell^2 \to \ell^2$ be defined by $T(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$ (right shift). Then $\|T\| = 1$.

Example 4. The multiplication operator $(M_g f)(x) = g(x)f(x)$ on $L^2$ is bounded with $\|M_g\| = \|g\|_\infty$.

The dual space of $X$ is $X^* = \mathcal{B}(X, \mathbb{R})$ (or $\mathcal{B}(X, \mathbb{C})$), the space of all bounded linear functionals.

Theorem 3.3. $X^*$ is always a Banach space.

Theorem 3.4. $(\ell^1)^* \cong \ell^\infty$ via $\varphi(x) = \sum x_n y_n$ for $y \in \ell^\infty$.

Theorem 3.5. $(\ell^p)^* \cong \ell^q$ for $1 \leq p < \infty$ where $1/p + 1/q = 1$.

Theorem 3.6. $(c_0)^* \cong \ell^1$ where $c_0 = \{(x_n) : x_n \to 0\}$.

Let $M$ be a subspace of a normed space $X$. The annihilator of $M$ is

$M^\perp = \{\varphi \in X^* : \varphi(x) = 0 \text{ for all } x \in M\}$

Let $N$ be a subspace of $X^*$. The pre-annihilator of $N$ is

${}^\perp N = \{x \in X : \varphi(x) = 0 \text{ for all } \varphi \in N\}$

Proposition 3.7. If $M \subseteq X$ is a subspace, then $M^\perp$ is a closed subspace of $X^*$. If $N \subseteq X^*$ is a subspace, then ${}^\perp N$ is a closed subspace of $X$.

Proposition 3.8. For a subspace $M \subseteq X$, ${}^\perp(M^\perp) = \overline{M}$ (the closure of $M$).

Proposition 3.9. For finite-dimensional subspaces $M \subseteq X$, $(X/M)^* \cong M^\perp$.

The double dual (or bidual) of $X$ is $X^{**} = (X^*)^*$. There is a natural embedding $J : X \hookrightarrow X^{**}$ defined by $(Jx)(\varphi) = \varphi(x)$ for $\varphi \in X^*$. The Hahn-Banach theorem guarantees that $J$ is an isometric embedding: $\|Jx\|_{X^{**}} = \|x\|_X$.

Definition. A normed space $X$ is reflexive if the canonical embedding $J : X \to X^{**}$ is surjective, i.e., $J(X) = X^{**}$.

Proposition 3.10. Every reflexive space is a Banach space.

Example. $\ell^p$ is reflexive for $1 < p < \infty$ since $(\ell^p)^{**} \cong (\ell^q)^* \cong \ell^p$.

Example. $L^p(\mu)$ is reflexive for $1 < p < \infty$.

Example. $\ell^1$, $\ell^\infty$, and $c_0$ are not reflexive. In particular, $(\ell^1)^{**} \cong (\ell^\infty)^* \supsetneq \ell^1$, and $(c_0)^{**} \cong \ell^\infty \supsetneq c_0$.

Theorem 3.11. A Banach space $X$ is reflexive if and only if its closed unit ball is weakly compact.

Theorem 3.12. If $X$ is reflexive, then every bounded sequence has a weakly convergent subsequence (Eberlein-Smulian theorem). In particular, every continuous linear functional achieves its norm on the closed unit ball.

Worked Example. Show that $c_0$ is not reflexive. Since $(c_0)^* \cong \ell^1$ by Theorem 3.6, and $(\ell^1)^* \cong \ell^\infty$ by Theorem 3.4, we have $(c_0)^{**} \cong \ell^\infty$. The canonical embedding $J : c_0 \hookrightarrow \ell^\infty$ is the inclusion map. Since $\ell^\infty$ contains bounded sequences that do not converge to zero (e.g., the constant sequence $(1, 1, 1, \ldots)$), $J$ is not surjective, so $c_0$ is not reflexive.

• Boundedness and continuity are equivalent for linear operators between normed spaces.
• The operator norm is submultiplicative: $\|ST\| \leq \|S\|\|T\|$ for composable operators.
• Reflexivity implies the Banach space property but not conversely.
• The double dual $X^{**}$ is always reflexive when $X$ is a Banach space.

• Assuming that every bounded linear functional on a subspace extends to the whole space without invoking the Hahn-Banach theorem. Extension requires the Hahn-Banach theorem and is not automatic.
• Confusing weak convergence with strong convergence. A sequence can converge weakly but not strongly (e.g., the standard basis in $\ell^2$).
• Forgetting that $\mathcal{B}(X,Y)$ is a Banach space only when $Y$ is complete. If $Y$ is not complete, the space of bounded operators need not be.
• Assuming reflexivity when only the canonical embedding is injective. Injectivity holds for all normed spaces by Hahn-Banach; reflexivity requires surjectivity.

• Quantum mechanics: Observables are modelled as self-adjoint bounded operators on Hilbert spaces.
• Numerical analysis: The spectral radius of an iteration matrix determines convergence of iterative methods.
• Partial differential equations: Bounded operators on Sobolev spaces encode weak formulations of PDEs.
• Signal processing: Bounded linear operators on $L^2$ spaces represent filters and transforms.

Theorem 3.13 (Open Mapping Theorem). If $X$ and $Y$ are Banach spaces and $T : X \to Y$ is a surjective bounded linear operator, then $T$ is an open map (it maps open sets to open sets).

Corollary 3.14 (Bounded Inverse Theorem). If $T : X \to Y$ is a bijective bounded linear operator between Banach spaces, then $T^{-1}$ is also bounded.

Theorem 3.15 (Closed Graph Theorem). A linear operator $T : X \to Y$ between Banach spaces is bounded if and only if its graph $\{(x, Tx) : x \in X\}$ is closed in $X \times Y$.

• Normed Spaces and Banach Spaces: Defines the domain and codomain spaces for bounded linear operators and establishes the dual space framework.

• The Fundamental Theorems: Presents the Hahn-Banach, open mapping, and closed graph theorems that govern the behaviour of bounded operators between Banach spaces.

• Compact Operators: Studies a subclass of bounded operators with finite-dimensional-like properties, crucial for spectral theory.

• Quantum Mechanics

• Graph Theory

• Classical Mechanics

• Electromagnetism

flowchart TD
    A[3_Bounded Linear Operators] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Bounded linear operators are the continuous linear maps between normed spaces — they do not blow up small inputs into large outputs. The operator norm measures the maximum stretching factor. The open mapping theorem says that surjective bounded operators between Banach spaces are automatically open maps, which implies the bounded inverse theorem: an invertible bounded operator has a bounded inverse. The closed graph theorem provides a practical test: a linear operator is bounded if and only if its graph is closed. These three results — open mapping, bounded inverse, and closed graph — are the fundamental theorems of functional analysis, guaranteeing that well-behaved operators behave as expected.

Problem. Let $T : C[0,1] \to C[0,1]$ be defined by $(Tf)(x) = xf'(x)$. Show that $T$ is unbounded.

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Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

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