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Local boundedness
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<h2 id="locally-bounded-function">Locally bounded function</h2> <p>A <a href="/facts/Real_number/R02gw5Pb">real-valued</a> or <a href="/facts/Complex_number/2w7mqI7e">complex-valued</a> function f {\displaystyle f} defined on some <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X {\displaystyle X} is called a locally bounded functional if for any x 0 ∈ X {\displaystyle x_{0}\in X} there exists a <a href="/facts/Neighborhood_(mathematics)/XHKgrpkp">neighborhood</a> A {\displaystyle A} of x 0 {\displaystyle x_{0}} such that f ( A ) {\displaystyle f(A)} is a <a href="/facts/Bounded_set/yCldjtoy">bounded set</a>. That is, for some number M > 0 {\displaystyle M>0} one has | f ( x ) | ≤ M for all x ∈ A . {\displaystyle |f(x)|\leq M\quad {\text{ for all }}x\in A.} </p><p>In other words, for each x {\displaystyle x} one can find a constant, depending on x , {\displaystyle x,} which is larger than all the values of the function in the neighborhood of x . {\displaystyle x.} Compare this with a <a href="/facts/Bounded_function/whVjbtuk">bounded function</a>, for which the constant does not depend on x . {\displaystyle x.} Obviously, if a function is bounded then it is locally bounded. The converse is not true in general (see below). </p><p>This definition can be extended to the case when f : X → Y {\displaystyle f:X\to Y} takes values in some <a href="/facts/Metric_space/gnBBRXtc">metric space</a> ( Y , d ) . {\displaystyle (Y,d).} Then the inequality above needs to be replaced with d ( f ( x ) , y ) ≤ M for all x ∈ A , {\displaystyle d(f(x),y)\leq M\quad {\text{ for all }}x\in A,} where y ∈ Y {\displaystyle y\in Y} is some point in the metric space. The choice of y {\displaystyle y} does not affect the definition; choosing a different y {\displaystyle y} will at most increase the constant r {\displaystyle r} for which this inequality is true. </p> <h2 id="examples">Examples</h2> <ul><li>The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = 1 x 2 + 1 {\displaystyle f(x)={\frac {1}{x^{2}+1}}} is bounded, because 0 ≤ f ( x ) ≤ 1 {\displaystyle 0\leq f(x)\leq 1} for all x . {\displaystyle x.} Therefore, it is also locally bounded.</li></ul> <ul><li>The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = 2 x + 3 {\displaystyle f(x)=2x+3} is not bounded, as it becomes arbitrarily large. However, it is locally bounded because for each a , {\displaystyle a,} | f ( x ) | ≤ M {\displaystyle |f(x)|\leq M} in the neighborhood ( a − 1 , a + 1 ) , {\displaystyle (a-1,a+1),} where M = 2 | a | + 5. {\displaystyle M=2|a|+5.} </li></ul> <ul><li>The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = { 1 x , if x ≠ 0 , 0 , if x = 0 {\displaystyle f(x)={\begin{cases}{\frac {1}{x}},&{\mbox{if }}x\neq 0,\\0,&{\mbox{if }}x=0\end{cases}}} is neither bounded nor locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.</li></ul> <ul><li>Any continuous function is locally bounded. Here is a proof for functions of a real variable. Let f : U → R {\displaystyle f:U\to \mathbb {R} } be continuous where U ⊆ R , {\displaystyle U\subseteq \mathbb {R} ,} and we will show that f {\displaystyle f} is locally bounded at a {\displaystyle a} for all a ∈ U {\displaystyle a\in U} Taking ε = 1 in the definition of continuity, there exists δ > 0 {\displaystyle \delta >0} such that | f ( x ) − f ( a ) | < 1 {\displaystyle |f(x)-f(a)|<1} for all x ∈ U {\displaystyle x\in U} with | x − a | < δ {\displaystyle |x-a|<\delta } . Now by the <a href="/facts/Triangle_inequality/KubLjkwr">triangle inequality</a>, | f ( x ) | = | f ( x ) − f ( a ) + f ( a ) | ≤ | f ( x ) − f ( a ) | + | f ( a ) | < 1 + | f ( a ) | , {\displaystyle |f(x)|=|f(x)-f(a)+f(a)|\leq |f(x)-f(a)|+|f(a)|<1+|f(a)|,} which means that f {\displaystyle f} is locally bounded at a {\displaystyle a} (taking M = 1 + | f ( a ) | {\displaystyle M=1+|f(a)|} and the neighborhood ( a − δ , a + δ ) {\displaystyle (a-\delta ,a+\delta )} ). This argument generalizes easily to when the domain of f {\displaystyle f} is any topological space.</li></ul> <ul><li>The converse of the above result is not true however; that is, a discontinuous function may be locally bounded. For example consider the function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } given by f ( 0 ) = 1 {\displaystyle f(0)=1} and f ( x ) = 0 {\displaystyle f(x)=0} for all x ≠ 0. {\displaystyle x\neq 0.} Then f {\displaystyle f} is discontinuous at 0 but f {\displaystyle f} is locally bounded; it is locally constant apart from at zero, where we can take M = 1 {\displaystyle M=1} and the neighborhood ( − 1 , 1 ) , {\displaystyle (-1,1),} for example.</li></ul> <h2 id="locally-bounded-family">Locally bounded family</h2> <p>A <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> (also called a <a href="/facts/Indexed_family/aFqfzJN0">family</a>) <i>U</i> of real-valued or complex-valued functions defined on some topological space X {\displaystyle X} is called locally bounded if for any x 0 ∈ X {\displaystyle x_{0}\in X} there exists a <a href="/facts/Neighborhood_(topology)/XHKgrpkp">neighborhood</a> A {\displaystyle A} of x 0 {\displaystyle x_{0}} and a positive number M > 0 {\displaystyle M>0} such that | f ( x ) | ≤ M {\displaystyle |f(x)|\leq M} for all x ∈ A {\displaystyle x\in A} and f ∈ U . {\displaystyle f\in U.} In other words, all the functions in the family must be locally bounded, and around each point they need to be bounded by the same constant. </p><p>This definition can also be extended to the case when the functions in the family <i>U</i> take values in some metric space, by again replacing the absolute value with the distance function. </p> <h2 id="examples">Examples</h2> <ul><li>The family of functions f n : R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} } f n ( x ) = x n {\displaystyle f_{n}(x)={\frac {x}{n}}} where n = 1 , 2 , … {\displaystyle n=1,2,\ldots } is locally bounded. Indeed, if x 0 {\displaystyle x_{0}} is a real number, one can choose the neighborhood A {\displaystyle A} to be the interval ( x 0 − a , x 0 + 1 ) . {\displaystyle \left(x_{0}-a,x_{0}+1\right).} Then for all x {\displaystyle x} in this interval and for all n ≥ 1 {\displaystyle n\geq 1} one has | f n ( x ) | ≤ M {\displaystyle |f_{n}(x)|\leq M} with M = 1 + | x 0 | . {\displaystyle M=1+|x_{0}|.} Moreover, the family is <a href="/facts/Uniformly_bounded/nyxwk4Ix">uniformly bounded</a>, because neither the neighborhood A {\displaystyle A} nor the constant M {\displaystyle M} depend on the index n . {\displaystyle n.} </li></ul> <ul><li>The family of functions f n : R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} } f n ( x ) = 1 x 2 + n 2 {\displaystyle f_{n}(x)={\frac {1}{x^{2}+n^{2}}}} is locally bounded, if n {\displaystyle n} is greater than zero. For any x 0 {\displaystyle x_{0}} one can choose the neighborhood A {\displaystyle A} to be R {\displaystyle \mathbb {R} } itself. Then we have | f n ( x ) | ≤ M {\displaystyle |f_{n}(x)|\leq M} with M = 1. {\displaystyle M=1.} Note that the value of M {\displaystyle M} does not depend on the choice of x0 or its neighborhood A . {\displaystyle A.} This family is then not only locally bounded, it is also uniformly bounded.</li></ul> <ul><li>The family of functions f n : R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} } f n ( x ) = x + n {\displaystyle f_{n}(x)=x+n} is not locally bounded. Indeed, for any x {\displaystyle x} the values f n ( x ) {\displaystyle f_{n}(x)} cannot be bounded as n {\displaystyle n} tends toward infinity.</li></ul> <h2 id="topological-vector-spaces">Topological vector spaces</h2> <p class="note">See also: <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">Bounded set (topological vector space)</a>, <a href="/facts/Normable_space/rmDljfyE">Normable space</a>, and <a href="/facts/Kolmogorov%27s_normability_criterion/ckaPRrE9">Kolmogorov's normability criterion</a></p> <p>Local boundedness may also refer to a property of <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a>, or of functions from a topological space into a topological vector space (TVS). </p> <h3>Locally bounded topological vector spaces</h3> <p class="note">Main article: <a href="/facts/Seminormed_space/vMuescKp">Seminormed space</a></p> <p>A <a href="/facts/Subset/SJGERmbA">subset</a> B ⊆ X {\displaystyle B\subseteq X} of a topological vector space (TVS) X {\displaystyle X} is called <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded</a> if for each neighborhood U {\displaystyle U} of the origin in X {\displaystyle X} there exists a real number s > 0 {\displaystyle s>0} such that B ⊆ t U for all t > s . {\displaystyle B\subseteq tU\quad {\text{ for all }}t>s.} A locally bounded TVS is a TVS that possesses a bounded neighborhood of the origin. By <a href="/facts/Kolmogorov%27s_normability_criterion/ckaPRrE9">Kolmogorov's normability criterion</a>, this is true of a locally convex space if and only if the topology of the TVS is induced by some <a href="/facts/Seminorm/vMuescKp">seminorm</a>. In particular, every locally bounded TVS is <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">pseudometrizable</a>. </p> <h3>Locally bounded functions</h3> <p>Let f : X → Y {\displaystyle f:X\to Y} a function between topological vector spaces is said to be a locally bounded function if every point of X {\displaystyle X} has a neighborhood whose <a href="/facts/Image_(mathematics)/KANefMAl">image</a> under f {\displaystyle f} is bounded. </p><p>The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces: </p> Theorem. A topological vector space X {\displaystyle X} is locally bounded if and only if the <a href="/facts/Identity_map/9AtsP0LC">identity map</a> id X : X → X {\displaystyle \operatorname {id} _{X}:X\to X} is locally bounded. <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bornological_space/kpl5B2Ez">Bornological space</a> – Space where bounded operators are continuous</li> <li><a href="/facts/Bounded_operator/LYmDTIqR">Bounded operator</a> – Linear transformation between topological vector spaces</li> <li><a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">Bounded set (topological vector space)</a> – Generalization of boundedness</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://planetmath.org/LocallyBounded">PlanetMath entry for Locally Bounded</a></li> <li><a href="https://ncatlab.org/nlab/show/locally+bounded+category">nLab entry for Locally Bounded Category</a></li></ul>