Chapter 2

Bounded Linear Operators

Boundedness is the analytic condition that makes an operator well-behaved — and, remarkably, it turns out to be exactly the same thing as continuity.

Linear operators

Let $H$ and $K$ be Hilbert spaces. A map $T : H \to K$ is a linear operator if $T(\alpha x + y) = \alpha T x + T y$ for all $x, y \in H$, $\alpha \in \mathbb{C}$. We focus mainly on operators $T : H \to H$ on a single Hilbert space.

Boundedness

Definition A linear operator $T : H \to H$ is bounded if there exists a constant $C \geq 0$ such that $$ \|Tx\| \leq C \|x\| \qquad \text{for all } x \in H. $$ The set of all bounded linear operators on $H$ is denoted $B(H)$ (also written $\mathcal{L}(H)$ or $\mathcal{B}(H)$ in the literature).

In infinite dimensions, boundedness is a genuine restriction: not every linear operator is bounded. A classical example is differentiation on a dense subspace of $L^2$, which is unbounded — it can amplify high-frequency components without limit.

Boundedness is equivalent to continuity

Theorem For a linear operator $T : H \to H$, the following are equivalent:
  1. $T$ is bounded.
  2. $T$ is continuous on $H$.
  3. $T$ is continuous at $0$.
  4. $T$ is continuous at some point $x_0 \in H$.

Sketch. (1) $\Rightarrow$ (2): if $\|Tx\| \leq C\|x\|$ then $\|Tx - Ty\| = \|T(x-y)\| \leq C\|x - y\|$, so $T$ is Lipschitz, hence continuous. (2) $\Rightarrow$ (3) $\Rightarrow$ (4) are immediate. For (4) $\Rightarrow$ (1): continuity at $x_0$ gives a $\delta$ such that $\|z\| \le \delta \Rightarrow \|Tz\| \le 1$ (using linearity, $T(x_0+z) - Tx_0 = Tz$); scaling any $x \neq 0$ to have norm $\delta$ and using linearity of $T$ then bounds $\|Tx\|$ by $\frac{1}{\delta}\|x\|$.

Why linearity matters here This equivalence relies crucially on linearity — a general continuous nonlinear map need not satisfy any global Lipschitz-type bound. For linear maps, local control at a single point propagates to a global bound everywhere.

$B(H)$ as an algebra

$B(H)$ is closed under addition, scalar multiplication, and composition: if $S, T \in B(H)$ then $S + T$, $\alpha T$, and $ST$ (meaning $S \circ T$) are all in $B(H)$. Composition is associative but not commutative in general, so $B(H)$ is a (typically noncommutative) algebra with identity $I$. The next chapter introduces the norm that makes $B(H)$ a Banach algebra.