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A set with an upper (respectively, lower) bound is said to be bounded from above or majorized [1] (respectively bounded from below or minorized) by that bound. The terms bounded above ( bounded below ) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
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 ...
The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space R n is compact if and only if it is closed and
Let : a function between topological vector spaces is said to be a locally bounded function if every point of has a neighborhood whose image under is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:
The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example (,) [,] = (,]. If R {\displaystyle \mathbb {R} } is viewed as a metric space , its open balls are the open bounded intervals ( c + r , c − r ) , and its closed balls ...
Each set has a supremum (infimum), if it is bounded from above (below). Proof: Without loss of generality one can look at a set A ⊂ R {\displaystyle A\subset \mathbb {R} } that has an upper bound. One can now construct a sequence ( I n ) n ∈ N {\displaystyle (I_{n})_{n\in \mathbb {N} }} of nested intervals I n = [ a n , b n ] {\displaystyle ...
A densely defined symmetric [clarification needed] operator T on a Hilbert space H is called bounded from below if T + a is a positive operator for some real number a. That is, Tx|x ≥ −a ||x|| 2 for all x in the domain of T (or alternatively Tx|x ≥ a ||x|| 2 since a is arbitrary). [8] If both T and −T are bounded from below then T is ...
Indeed, the elements of define a pointwise bounded family of continuous linear forms on the Banach space := ′, which is the continuous dual space of . By the uniform boundedness principle, the norms of elements of S , {\displaystyle S,} as functionals on X , {\displaystyle X,} that is, norms in the second dual Y ″ , {\displaystyle Y'',} are ...