58 Transactions of the Academy of Science of St. Louis 
system, called the norm. ; 
P.11. If there is an element whose norm is zero, it is unique. 
P.12. [&-x|| - | &| - ||| 
P.13. /|x+-y\] <||x\| +|]y| 
It is possible to show that the postulates are consistent, and 
that P.2 and P.9 may be proved from the remaining ones, which 
are independent among themselves. 
The element (—x) is defined as a) ey. ; 
Neighborhood. The set of points x, which satisfy the in- 
equality 
liz — wll <p >0, 
where x is a given element, and p a given positive number, is 
called the p neighborhood of x». The element Xp is called the 
center of the neighborhood. 
Region. If every point of a given set R is the center of a 
neighborhood which belongs to the set, then R is called a region. 
Connected region. Let x) and x, be any two points of a re- 
gion R. If there exists a chain of neighborhoods, finite in num- 
ber, all contained in R, and such that each neighborhood con- 
tains the center of the succeeding one, with x» being the first 
center and x, the last, then R is said to be connected. The set 
of points xo, - - - , x; that serve as centers for the neighborhoods 
of the chain is called the route of the chain. 
Sim ply-connected region. Let x) and x, be any two points 
of a connected region R,and C and C’ any two chains that ex- 
tend from x9 to x. Then if we can replace C by a succession 
of chains such that the neighborhoods of each chain contain the 
in, and if after a finite number of steps 
’, R is said to be simply-connected. 
Limit point. A set of points Ep of E is said to have a limit 
point xo if it contains points different from x, in every p neigh- 
borhood of xp. 
Convergence. 
An infinite sequence of elements {xn} is said 
to converge to a limit xo if for every p greater than zero, there 
corresponds an integer ”,, such that for all » > i, | Xn —Xo| <p. 
Completeness. We now add another postulate so that the 
Space considered becomes a complete linear vector, or Banach, 
space. . 
P.14. In order that a sequence { <a) converges, it is suffi- 
cient that for every p>0 there exists an integer , such that for 
all x>n, and all p>0, : 
