Page 326 - Advanced Linear Algebra
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310 Advanced Linear Algebra
Continuity
Continuity plays a central role in the study of linear operators on infinite-
dimensional inner product spaces.
Definition Let ¢ 4 ¦ 4 Z be a function from the metric space ²4Á ³ to the
Z
Z
metric space ²4 Á ³ . We say that is continuous at % 4 if for any ,
there exists a such that
²%Á % ³ ¬ ² ²%³Á ²% ³³ Z
or, equivalently,
)²% Á ³ )² ²% ³Á ³ 5
4
(See Figure 12.2. ) A function is continuous if it is continuous at every
% 4.
Figure 12.2
We can use the notion of convergence to characterize continuity for functions
between metric spaces.
Theorem 12.4 A function ¢ 4 ¦ 4 Z is continuous if and only if whenever
²% ³ is a sequence in 4 that converges to % 4, then the sequence ² ²% ³³
converges to ²% ³ , in short,
²% ³ ¦ % ¬ ² ²% ³³ ¦ ²% ³
Proof. Suppose first that is continuous at % and let % ² ³ ¦ % . Then, given
, the continuity of implies the existence of a such that
²)²% Á ³³ )² ²% ³Á ³
Since ²% ³¦% , there exists an 5 such that % )²% Á ³ for 5 and
so
5 ¬ ²% ³ )² ²% ³Á ³
Thus, ²% ³ ¦ ²% ³ .
Conversely, suppose that ²% ³¦% implies ² ²% ³³¦ ²% ³ . Suppose, for the
purposes of contradiction, that is not continuous at % . Then there exists an
