An important concept in Functional Analysis, Personal spaces were first developed by the famed mathemagician Big Norman, though it is now credited to Cauchy who has proved using Proof by intimidation that it was actually all done by him wearing a different hat.
As any slack-jawed yokel knows, Normed Spaces are spaces of thingies equipped with a John Norman novel, or 'Norm'. What fewer slack-jawed yokels know is that this simple definition rarely leads to a Normed space with interesting properties. For example, under this definition the Empty Set, equipped with any fairly primitive norm (such as the Ninety-Five Chapters On How The Blacksmiths Of Gor Make Swords norm), counts as a Normed Space even though it's of no practical use whatsoever, much like Celine Dion.
The problem of distinguishing between an interesting Normed Space (such as the famed Scantily-Clad Women Of Gor Bouncing Up And Down Repeatedly Space) and a dull one (such as the stupid, stupid space mentioned earlier) was still stunting the growth of Functional Analysis up until 1903. It was then that mathemagician Big Norman developed the subcategory of Normed spaces known as Personal spaces.
A Normed space X is said to be a Personal space if:
What does this mean? Well, for one thing it meant that Big Norman finally got laid, and how. But on a less important level, it means that no element of a Personal space X can ever be far enough away from another to have a nice breather. Under this constraint, the more timid and retiring norms are unable to adequately describe the hot steamy element-on-element action.
The Fundamental Theorem of Personal spaces
Indeed, Big Norman was able to develop the greatest theorem of Personal spaces, aptly named Cauchy's Personal Space Theorem. This theorem proves decisively that any Personal space must be described by a norm with a raunchiness index (or Cauchy Index) above 2.5 Brittneys. A proof is yet to arrive from Cauchy, but he has assured the international community of mathemagicians that it's really easy, and "you would finish it in a snap if you were as good as I am."
Corollaries of the Fundamental Theorem of Personal spaces
These corrollaries of the Fundamental Theorem of Personal spaces are stated without proof, because if you need help proving them then you're a big stupid-head.
Corollary 1 (The Cobbler Factor): For a personal space .
Corolary 2: In Colloray 1, the constant r satisifies .
In common language, as might be used by a navvie or cutpurse, this corollorollies state that "Eee, a person needs more personal space if they've had too much PI, loike. The more PI = the more personal space is squared = the more rounded that them person becomes, chuck."
1. Let X be the Normed space spanned by , for some t belonging to the standard Telly Set T. Describe a norm such that . Why is X not a Personal space?
2. Let x be an element of the Personal space of functions on the Cauchy Plane, such that x is not a 'straight function' (i.e. bihomomorphic or homomorphic). Prove that x is the zero function. Hint: Remember that x must be acted on by a sufficiently raunchy norm.
3. Prove Cauchy's Personal Space Theorem. You may assume the consequences of de Niro's You Talkin' To Me Lemma, provided you state them clearly and in a loud, confident voice. Hint: If you manage this, please get in touch, because we're really, really up the creek on this one.