One more case merits mentioning here.
One reason for using this rather elaborate model is that it permits a simple and concrete definition of the realization.
One simple way to get a Banach algebra which is not amenable is to take ......
Our definition agrees with the one in .
The cases $p=1$ and $p=2$ will be the ones of interest to us.
Consider the differences between these integrals and the corresponding ones with $f$ replaced by $g$.
It has some basic properties in common with another most important class of functions, namely, the continuous ones.
The geodesics (8) are the only ones that realize the distance between their endpoints.
Then the one and only integral curve of $L$ starting from $x$ is the straight line $l$.
Then $G$ has ten normal subgroups and as many non-normal ones.
The first two are simpler than the third one. [Or: the third; not: “The first two ones”]
Now, $F$ has many points of continuity. Suppose $x$ is one.
Each of the functions on the right of (9) is one to which our theorem applies.
A principal ideal is one that is generated by a single element.
If $f$ is an $n$-simplex in $U$, then $f'$ is one in $V$.
Suppose that of all such solutions, $(x,y,z)$ is one with $y$ minimal.
In addition to a contribution to $W_1$, there may also be one to $W_2$.
Necessarily, one of $X$ and $Y$ is in $Z$.
Here the interesting questions are not about individual examples, but about the asymptotic behaviour of the set of examples as one or another of the invariants (such as the genus) goes to infinity.
Instead of dealing with lines one by one, we deal with collections of lines simultaneously.
Clearly, $A$ is thereby put in one-to-one correspondence with $B$.
The algorithm examines only roughly one-quarter to one-third of the characters.
Asymptotically, more than one-fifth of the polynomials $B_n(x)$ are irreducible.
The other player is one-third as fast.
[The numbers 1 to 12, when used for counting objects (without units of measurement), should be written in words: There exists exactly one such map; for other uses, sometimes figures permit avoiding ambiguity, e.g. it is 1 less than the other.]