Bounded Linear Operators — Mathematics Knowledge System

Overview

A bounded linear operator between normed spaces is a linear map T: X -> Y for which there exists a constant C such that ||Tx|| <= C||x|| for all x. Boundedness is equivalent to continuity for linear maps between normed spaces. The collection B(X,Y) of all bounded linear operators is itself a normed space (and a Banach space when Y is complete). The dual space X* = B(X,F) is the space of bounded linear functionals and plays a central role in the duality theory of normed spaces.

Intuition

A linear operator T is bounded if it does not 'stretch' vectors by an arbitrarily large factor. A continuous function on a metric space cannot send nearby inputs to wildly separated outputs; for linear maps, this is equivalent to not stretching norms beyond a fixed multiple. The operator norm ||T|| = sup_{||x||=1} ||Tx|| measures the maximum stretching factor. The dual space X* captures all 'measurements' one can make of a vector x: each functional f in X* gives a scalar f(x), and the collection of all such measurements encodes the geometry of X.

Formal Definition

Definition

Let X and Y be normed vector spaces. A linear map T: X -> Y is bounded if there exists C >= 0 such that ||Tx||_Y <= C||x||_X for all x in X. The operator norm is ||T|| = sup_{x!=0} ||Tx||/||x|| = sup_{||x||<=1} ||Tx||. The space B(X,Y) of all bounded linear operators is a normed space under this norm, and a Banach space when Y is complete. The dual space is X* = B(X,F).

\|T\| = \sup_{x \ne 0} \frac{\|Tx\|_Y}{\|x\|_X} = \sup_{\|x\|=1}\|Tx\|_Y

Operator norm

\|Tx\| \le \|T\|\,\|x\| \quad \forall\, x \in X

Boundedness estimate

\|ST\| \le \|S\|\,\|T\|

Sub-multiplicativity

X^* = \mathcal{B}(X, \mathbb{F}),\quad \|f\|_{X^*} = \sup_{\|x\|\le 1}|f(x)|

Dual space

Notation

Notation	Meaning
$\mathcal{B}(X,Y)$	Bounded linear operators from X to Y
$\mathcal{B}(X)$	Bounded linear operators from X to X (operator algebra)
$X^*$	Dual space of X: bounded linear functionals on X
$T^*$	Adjoint or dual operator of T

Properties

Operator norm is sub-multiplicative

For T in B(X,Y) and S in B(Y,Z): \|S \circ T\| \le \|S\|\,\|T\|.

Dual operator

For T in B(X,Y) the adjoint T^*: Y^* -> X^* defined by (T^* g)(x) = g(Tx) satisfies \|T^*\| = \|T\|.

Theorems

Theorem 4.1: Continuity Equivalence for Linear Maps

Let T: X -> Y be a linear map between normed spaces. The following are equivalent: (i) T is bounded; (ii) T is continuous; (iii) T is continuous at 0; (iv) T is uniformly continuous.

Theorem 4.2: B(X,Y) is Banach when Y is Banach

If X is a normed space and Y is a Banach space, then B(X,Y) equipped with the operator norm is a Banach space. In particular, the dual X* of any normed space X is always a Banach space.

Theorem 4.3: Natural Isometric Embedding into Bidual

The canonical map J: X -> X**, defined by (Jx)(f) = f(x) for f in X*, is an isometric linear injection. Thus every normed space embeds isometrically into its bidual.

Worked Examples

1
L is linear. Compute: ||Lx||_2^2 = sum_{n>=2} x_n^2 <= sum_{n>=1} x_n^2 = ||x||_2^2. So ||L|| <= 1.
2
Take x = e_2 = (0,1,0,...). Then Le_2 = e_1 = (1,0,...), so ||Le_2||_2 = 1 = ||e_2||_2.
3
Thus ||L|| = 1.

✓ Answer

The operator norm of the left-shift is ||L|| = 1.

Practice Problems

Mediumproof writing

Prove that the dual space X* of any normed space X is always a Banach space, even if X itself is not complete.

Mediumfree response

Give an example of an unbounded linear operator on an infinite-dimensional normed space.

MediumMultiple choice

Which statement correctly characterises bounded linear operators?

Bounded linear operators are the same as continuous linear operatorsBounded implies continuous but not converselyContinuous implies bounded but not converselyBounded and continuous are unrelated for linear maps

Common Mistakes

Common Mistake

Assuming that every linear operator is bounded.

In infinite dimensions, unbounded linear operators exist (e.g., differentiation on polynomials). Boundedness is a real restriction that must be verified.

Common Mistake

Confusing the operator norm with the spectral radius.

The operator norm ||T|| and the spectral radius r(T) = lim ||T^n||^{1/n} are different. For self-adjoint operators on Hilbert spaces, r(T) = ||T||, but this is a special theorem.

Quiz

Historical Background

The study of linear operators on function spaces grew from the theory of integral equations in the early twentieth century. Fredholm (1900) and Hilbert (1904-1910) studied what we now call bounded operators on L^2. The operator norm was made explicit by Riesz in 1913. Banach's 1932 monograph gave the definitive treatment of bounded operators between general Banach spaces, including the three cornerstone theorems (open mapping, closed graph, uniform boundedness).

1900
Fredholm studies linear integral operators
Ivar Fredholm
1910
Hilbert develops spectral theory for symmetric operators on l^2
David Hilbert
1913
Riesz defines and studies the operator norm
Frédéric Riesz
1932
Banach's monograph unifies bounded operator theory
Stefan Banach

Summary

A bounded linear operator T: X -> Y satisfies ||Tx|| <= C||x|| for some C >= 0; equivalently, T is continuous.
The operator norm ||T|| = sup_{||x||=1} ||Tx|| is the smallest valid bounding constant.
B(X,Y) is a Banach space when Y is Banach; in particular, the dual X* is always Banach.
The canonical embedding J: X -> X** is an isometric injection; if surjective, X is called reflexive.
The adjoint T*: Y* -> X* is defined by (T*g)(x) = g(Tx) and satisfies ||T*|| = ||T||.

References

BookRudin, W. — Functional Analysis (2nd ed.), Chapter 2
BookConway, J. — A Course in Functional Analysis, Chapter 3
Websiteen.wikipedia.org