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A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...
As particular examples of Banach spaces, Dunford & Schwartz (1958, Chapter IV) consider spaces of sequences of bounded variation, in addition to the spaces of functions of bounded variation. The total variation of a sequence x = ( x i ) of real or complex numbers is defined by
Every uniformly convergent sequence of bounded functions is uniformly bounded. The family of functions f n ( x ) = sin n x {\displaystyle f_{n}(x)=\sin nx} defined for real x {\displaystyle x} with n {\displaystyle n} traveling through the integers , is uniformly bounded by 1.
Corollary — If a sequence of bounded operators () converges pointwise, that is, the limit of (()) exists for all , then these pointwise limits define a bounded linear operator . The above corollary does not claim that T n {\displaystyle T_{n}} converges to T {\displaystyle T} in operator norm, that is, uniformly on bounded sets.
On the other hand, for =, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set. The Mandelbrot set can also be defined as the connectedness locus of the family of quadratic polynomials f ( z ) = z 2 + c {\displaystyle f(z)=z^{2}+c} , the subset of the space of parameters c {\displaystyle c} for which the ...
This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition.
Definition: A set is sequentially compact if every sequence {} in has a convergent subsequence converging to an element of . Theorem: A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}} is sequentially compact if and only if A {\displaystyle A} is closed and bounded.
Equivalently, a bounded operator T is compact if, for any bounded sequence {x k}, the sequence {Tx k} has a convergent subsequence. Many integral operators are compact, and in fact define a special class of operators known as Hilbert–Schmidt operators that are especially important in the study of integral equations.