Bounded
The term bounded appears in different parts of mathematics where a notion of "size" can be given. The basic intuitive meaning common to all of them is that something is of finite size, and that this is the case if it is smaller than some other object that has a finite size. (Otherwise it is unbounded.) For the precise definition no precise definition of 'size' is needed.
Calculus
A set S of real numbers is called bounded above if there is a real number k such that k > s for all s in S. The number k is called an upper bound of S. The terms bounded below and lower bound are similarly defined. A set S is bounded if it is bounded both above and below. Therefore, a set is bounded if it is contained in a finite interval.
A function f : X -> R is bounded on X if its image f(X) is a bounded subset of R.
Metric spaces
A set S in a metric space (M, d) is bounded if it is contained in a ball of finite radius, i.e. if there exists x in M and r > 0 such that for all s in S, we have d(x, s) < r.
Functional analysis
A set S in a topological vector space is bounded if it is contained in some multiple of every basic neighbourhood of zero. A bounded linear operator is continuous.
That--the world.html">world watched o'er.
Haughty plants within that humble.html">humble bed
With unblushing ribaldry besets!
Suffer now his ev'ry nauseous jest,
And your heavy slumber shake away!
Hearken to the trump's exulting song!
Rouse ye, then, ye kings!
Seven sleepers!--to the clarion hark!
Shouts from out a thousand barrels whizz;
Soon the bristly boar rolls in his blood,--
Tow'rd me scornful echoes ninefold come,
Sleep assails the page by slow degrees,
Of--her sleeping-room.
Not an answer--hushed and still is all--
Which enshrouds their humble flatt'rers glance?
Since the blind jade, Fate, a world has thrust
Marshalled in your proudly childish groups,
Though the sound may please the vulgar ear,
Powers so great, but mean.
Let your towering shame be hid from sight
From the ambush of the throne outspring!
Through the purple, vengeance will, ere long,
THE SATYR AND MY MUSE.
An aged satyr sought
Attempting to pay court,
By Luna's pallid light,
Crept the unhappy sharp-eared wight;
And warbled many a lay,
And fiercely scraped away
As large as nuts, or larger;
Like music from Silenus' charger.
The Muse sat still, and played
And peevishly surveyed
Thou ugly, dirty dunce?
As Midas was Apollo once?
"Speak out, old horned boor
Thou'rt swarthy as a Moor,
In far Teutonia's land;
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On
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