Editing Bounded operator (section)

==Examples==

<ul>
<li>Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed [[matrix (mathematics)|matrix]].</li>
<li>Any linear operator defined on a finite-dimensional normed space is bounded.</li>
<li>On the [[sequence space]] <math>c_{00}</math> of eventually zero sequences of real numbers, considered with the <math>\ell^1</math> norm, the linear operator to the real numbers which returns the sum of a sequence is bounded, with operator norm 1. If the same space is considered with the <math>\ell^{\infty}</math> norm, the same operator is not bounded.</li>
<li>Many [[integral transform]]s are bounded linear operators. For instance, if 
<math display=block>K : [a, b] \times [c, d] \to \R</math>
is a continuous function, then the operator <math>L</math> defined on the space <math>C[a, b]</math> of continuous functions on <math>[a, b]</math> endowed with the [[uniform norm]] and with values in the space <math>C[c, d]</math> with <math>L</math> given by the formula
<math display=block>(Lf)(y) = \int_a^b\!K(x, y)f(x)\,dx, </math>
is bounded. This operator is in fact a [[compact operator]]. The compact operators form an important class of bounded operators.</li>
<li>The [[Laplace operator]] 
<math display=block>\Delta : H^2(\R^n) \to L^2(\R^n) \,</math>
(its [[domain of a function|domain]] is a [[Sobolev space]] and it takes values in a space of [[square-integrable function]]s) is bounded.</li>
<li>The [[shift operator]] on the [[Lp space]] <math>\ell^2</math> of all [[sequence]]s <math>\left(x_0, x_1, x_2, \ldots\right)</math> of real numbers with <math>x_0^2 + x_1^2 + x_2^2 + \cdots < \infty, \,</math> 
<math display=block>L(x_0, x_1, x_2, \dots) = \left(0, x_0, x_1, x_2, \ldots\right) </math> 
is bounded. Its operator norm is easily seen to be <math>1.</math></li>
</ul>

===Unbounded linear operators===

Let <math>X</math> be the space of all [[trigonometric polynomial]]s on <math>[-\pi, \pi],</math> with the norm 

<math display=block>\|P\| = \int_{-\pi}^{\pi}\!|P(x)|\,dx.</math>

The operator <math>L : X \to X</math> that maps a polynomial to its [[derivative]] is not bounded. Indeed, for <math>v_n = e^{i n x}</math> with <math>n = 1, 2, \ldots,</math> we have <math>\|v_n\| = 2\pi,</math> while <math>\|L(v_n)\| = 2 \pi n \to \infty</math> as <math>n \to \infty,</math> so <math>L</math> is not bounded.

===Properties of the space of bounded linear operators===
The space of all bounded linear operators from <math>X</math> to <math>Y</math> is denoted by <math>B(X, Y)</math>.
* <math>B(X, Y)</math> is a normed vector space.
* If <math>Y</math> is Banach, then so is <math>B(X, Y)</math>; in particular, [[dual space]]s are Banach.
* For any <math>A \in B(X, Y)</math> the kernel of <math>A</math> is a closed linear subspace of <math>X</math>.
* If <math>B(X, Y)</math> is Banach and <math>X</math> is nontrivial, then <math>Y</math> is Banach.