What is Bounded: Definition and 536 Discussions

In functional analysis, a bounded linear operator is a linear transformation



L
:
X

Y


{\displaystyle L:X\to Y}
between topological vector spaces (TVSs)



X


{\displaystyle X}
and



Y


{\displaystyle Y}
that maps bounded subsets of



X


{\displaystyle X}
to bounded subsets of



Y
.


{\displaystyle Y.}

If



X


{\displaystyle X}
and



Y


{\displaystyle Y}
are normed vector spaces (a special type of TVS), then



L


{\displaystyle L}
is bounded if and only if there exists some



M
>
0


{\displaystyle M>0}
such that for all



x


{\displaystyle x}
in



X
,


{\displaystyle X,}


The smallest such



M
,


{\displaystyle M,}
denoted by




L

,


{\displaystyle \|L\|,}
is called the operator norm of



L
.


{\displaystyle L.}

A linear operator that is sequentially continuous or continuous is a bounded operator and moreover, a linear operator between normed spaces is bounded if and only if it is continuous.
However, a bounded linear operator between more general topological vector spaces is not necessarily continuous.

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  1. brotherbobby

    To find the boundedness of a given function

    ##\small{\texttt{(I could solve the for the upper limit explicitly.}}## ##\small{\texttt{However, not the same for the lower limit, except via inspection.)}}## I copy and paste the the problem as it appeared in the text. ##\rm(I)## : ##\texttt{The domain :}## The domain of the function is...
  2. C

    Proof that T is bounded below with ##inf T = 2M##

    My first solution is Let ##S = \{x_1, x_2, x_3, ..., x_n\}## ##T = \{2x_1, 2x_2, 2x_3, ... 2x_n\}## ##T = 2S## Therefore, ##inf T = inf 2S = 2inf S = 2M## May someone please know whether this counts as a proof? My second solution is, ##x ≥ M## ##2x ≥ 2M## ##y ≥ 2M## (Letting y = 2M) Let...
  3. Euge

    POTW A Nonlinear Elliptic PDE on a Bounded Domain

    Let ##D## be a smooth, bounded domain in ##\mathbb{R}^n## and ##f : D \to (0, \infty)## a continuous function. Prove that there exists no ##C^2##-solution ##u## of the nonlinear elliptic problem ##\Delta u^2 = f## in ##D##, ##u = 0## on ##\partial D##.
  4. chwala

    Determine if the given set is Bounded- Complex Numbers

    My interest is only on part (a). Wah! been going round circles to try understand why the radius = ##2##. I know that the given sequence is both bounded and monotonic. I can state that its bounded above by ##1## and bounded below by ##0##. Now when it comes to the radius=##2##, i can also say...
  5. J

    Bounded non-decreasing sequence is convergent

    So far this is what I have. Proof: Let p1, p2, p3 be a non-decreasing sequence. Assume that not all points of the sequence p1,p2,p3,... are equal. If the sequence p1,p2,p3,... converges to x then for every open interval S containing x there is a positive integer N s.t. if n is a positive integer...
  6. C

    I How can I use the concept of a proper map to show that a set is bounded?

    Dear Everybody, I am having some trouble with proving this set ##S=\{(x,y)\in \mathbb{R}^2: 3x^2-4xy+5y^2 \leq 5\}## is bounded. Find a real number ##R>0## such that ##\sqrt{x^2+y^2}\leq ## for all ##(x,y)\in S.## My attempt: ##3x^2-4xy+5y^2 =3x^2+(x-y)^2-(x+y)^2+5y^2 \\ \leq...
  7. C

    I Proving a convergent sequence is bounded

    Dear Everybody, I have a quick question about the \M\ in this proof: Suppose \b_n\ is in \\mathbb{R}\ such that \lim b_n=3\. Then, there is an \ N\in \mathbb{N}\ such that for all \n\geq\, we have \|b_n-3|<1\. Let M1=4 and note that for n\geq N, we have |b_n|=|b_n-3+3|\leq |b_n-3|+|3|<1+3=M1...
  8. Vividly

    B Understanding about Sequences and Series

    Homework Statement:: Tell me if a sequence or series diverges or converges Relevant Equations:: Geometric series, Telescoping series, Sequences. If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too? Also if I...
  9. chwala

    Bounded and monotonic sequences - Convergence

    I would like some clarity on the highlighted part. My question is, consider the the attached example ##(c)##, This sequence converges ( by using L'Hopital's rule)...now my question is, the sequence is indicated on text as not being monotonic...very clear. Does it imply that if a sequence is not...
  10. A

    Is there a way to prove that a set is bounded using calculus techniques?

    I know that for a set to be bounded it is bounded above and below, for the bound below is it 0 and n cannot equal 1 and u paper bound is inf but how do I prove that it is bounded?
  11. docnet

    Prove that |f| is bounded by a quotient

    I just spent 15 minutes to re-type all the Latex again because I lost everything while editing. why does this happen?? This is a huge waste of time. ##f## is entire so ##f## is holomorphic on ##\mathbb{D}∪C##. Also, ##\mathbb{D}∪C## is a connected set. By the maximum principle, ##f## restricted...
  12. H

    Matrix with a bounded mapping as an entry is bounded

    In a previous exercise I have shown that for a $$C^{*} algebra \ \mathcal{A}$$ which may or may not have a unit the map $$L_{x} : \mathcal{A} \rightarrow \mathcal{A}, \ L_{x}(y)=xy$$ is bounded. I.e. $$||L_{x}||_{\infty} \leq ||x||_{1}$$, $$x=(a, \lambda) \in \mathcal{\hat{A}} = \mathcal{A}...
  13. H

    Bounded operators on Hilbert spaces

    I have to show that for two bounded operators on Hilbert spaces ##H,K##, i.e. ##T \in B(H)## and ##S \in B(K)## that the formula ##(T \bigoplus S) (\alpha, \gamma) = (T \alpha, S \gamma)##, defined by the linear map ##T \bigoplus S: H \bigoplus K \rightarrow H \bigoplus K ## is bounded...
  14. Eclair_de_XII

    B Question about set containing subsequential limits of bounded sequence

    Let ##L\in E##. By definition, there is a subsequence ##\{x_{n_k}\}_{k\in\mathbb{N}}## that converges to ##L##. There is a natural number ##N## s.t. if ##n_k\geq N##, ##L\in(x_{n_k}-1,x_{n_k}+1)\subset(\inf\{x_n\}-1,\sup\{x_n\}+1)##. Hence, ##E## is a bounded set. If ##E## is a finite set, then...
  15. I

    A Bounded Packet Motion: Unravel the Asymptotically Helical Trajectory

    In the article 'Cellular vacuum' (Int. J. Theor. Phys. 21: 537-551, 1982), Minsky writes: "One can prove that any bounded packet which moves within a regular lattice must have an asymptotically helical trajectory.. . " He does not explain this statement further, nor does he give any references...
  16. L

    MHB Is non continuous function also not Bounded ?

    Dear all, I am trying to figure out if a non continuous function is also not bounded. I know that a continuous function in an interval, closed interval, is also bounded. Is a non continuous function in a closed interval not bounded ? I think not, it makes no sense. How do you prove it ? Thank...
  17. C

    Stuck at proving a bounded above Subsequence

    Summary:: x Let ## \{ a_{n} \} ## be a sequence. Prove: If for all ## N \in { \bf{N} } ## there exists ## n> N ## such that ## a_{n} \leq L ## , then there exists a subsequence ## \{ a_{n_{k}} \} ## such that ## a_{n_{k}} \leq L ## My attempt: Suppose that for all ## N \in {\bf{N}} ##...
  18. anemone

    MHB Area of the bounded regions between a straight line and a polynomial

    Let $P$ be a real polynomial of degree five. Assume that the graph of $P$ has three inflection points lying on a straight line. Calculate the ratios of the areas of the bounded regions between this line and the graph of the polynomial $P$.
  19. agnimusayoti

    Volume in the first octant bounded by the coordinate planes and x + 2y + z = 4.

    First, I try to make a sketch and from that I take limit of integration from: 1. ##z_1 = 0## to ##z_2 = 4 - x -2y## 2. ##x_1 = 0## to## x_2 = 4- 2y ## 3. ##y_1 = 0## to ##y_2 = 2## Then, I define infinitesimal volume element in the first octant as ##dV = 1/8 dz dz dy##. Therefore, $$V=1/8...
  20. F

    I Any surface bounded by the same curve in Stokes' theorem

    In Stokes' theorem, the closed line integral of f=the surface integral of curl f on ANY surface bounded by the same curve. But in Gauss' theorem, the surface integral of f on a surface=the volume integral of div f on a unique volume bounded by the surface. A surface can only enclose 1 volume...
  21. N

    Find the infimum and/or supremum and see if the set is bounded

    ##S_3 = \left\{ \ x∈ℝ : x^2+x+1≥0 \right\}## I am not sure if I have done this correctly. The infimum/supremum and maximum/minimum are confusing me a bit. This is how I started: ##x^2+x+1=0## ##x^2+x+ \frac1 4\ =\frac{-3} {4}\ ## ## \left\{ x^2+\frac 1 2\ \right\} ^2 +\frac 3 4\ = 0##...
  22. S

    MHB Maximizing Probability of Finding Explosive Device in Bounded Domain

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  23. Decimal

    Ambipolar diffusion and sheaths in a bounded plasma

    Hello, I am currently working through an introductory textbook on plasma physics, and I have encountered two topics that I separately understand but seem to be at odds with one another. In a quasi neutral plasma in steady state, the following relation must hold, $$\Gamma_i = \Gamma_e.$$ In...
  24. DaTario

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  25. F

    Is the set open, closed, neither, bounded, connected?

    Let ##z = a + bi##. Using the definition of modulus, we have ##\vert z - 3 \vert < 2## is equivalent to ##\sqrt{(a+3)^2 + b^2} < 2##. Squaring both sides we get ##(a+3)^2 + b^2 < 4##. This is the open disk center at ##3## with radius ##4## which we write as ##D[-3, 2]##. First we show...
  26. Math Amateur

    MHB Closed and Bounded Intervals are Compact .... Sohrab, Propostion 4.1.9 .... ....

    Closed and Bounded Intervals are Compact ... Sohrab, Proposition 4.1.9 ... ... I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Proposition...
  27. Math Amateur

    MHB Norm bounded Sets .... remarks by Garling in Section 11.2 Normed Spaces ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help in order to understand some...
  28. Math Amateur

    MHB Bounded in Norm .... Garling, Section 11.2: Normed Spaces ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help with some remarks by Garling concerning a...
  29. Math Amateur

    I Bounded in Norm .... Garling, Section 11.2: Normed Spaces ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help with some remarks by Garling concerning a subset...
  30. Drioton

    How do I find the area of the region bounded by following?

    Using integrals, consider the 7 requirements: Any my attempted solution that I have no idea where I am going: And the other one provides the graph:
  31. S

    Area of a bounded region using integration

    In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...
  32. JD_PM

    I Understanding why ##(y_n)_n## is a bounded sequence

    Suppose ##(y_n)_n## is a sequence in ##\mathbb{C}## with the following property: for each sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, also the series ##\sum_n \left(x_ny_n\right)## converges absolutely. Can you then conclude that ##(y_n)_n##...
  33. Jozefina Gramatikova

    Classical mechanics: Square well with Bounded particle

    My question is can we have negative energy in classical mechanics? Also I would need help for finding the velocity in part b)
  34. L

    A Same open sets + same bounded sets => same Cauchy sequences?

    Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...
  35. A

    MHB Bounds of the difference of a bounded band-limited function

    for continues signal (function) we have Bernstein inequality : $$ |{df(t)}/dt| \le 2AB\pi $$ where A=sup$|f(t)|$ and B is Bandwidth f(t), the question is:Is there a relationship for discrete function x[n] like this? $$ |x[n] -x[n-1] | \le\ \mu\ W $$ where $$ X[k] = \sum\limits_{k =...
  36. K

    Volume of revolution, region bounded by two functions

    Homework Statement Let R be the area in the xy-plane in the 1st quadrant which is bounded by the curves y^2+x^2 = 5, y = 2x and x = 0. (y-axis). Let T be the volume of revolution that appears when R is rotated around the Y axis. Find the volume of T. Homework EquationsThe Attempt at a Solution...
  37. J

    MHB Properties of Functions of Bounded Variation

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  38. J

    MHB Bounded Variation - Difference of Functions

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  39. T

    How to determine the volume of a region bounded by planes?

    Homework Statement Let G be the region bounded by the planes x=0,y=0,z=0,x+y=1and z=x+y. Homework Equations (a) Find the volume of G by integration. (b) If the region is a solid of uniform density, use triple integration to find its center of mass. The Attempt at a Solution [/B] My...
  40. evinda

    MHB Is this the desired bounded set of the wave equation?

    Hello! (Wave) I want to show for the initial value problem of the wave equation $$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$ that if the data (i.e. the initial data and the non-homogeneous term $f$) have compact support, then, at each time, the solution has also compact support. I...
  41. T

    Metric space of continuous & bounded functions is complete?

    Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
  42. J

    Decide if the energy surfaces in phase space are bounded

    Homework Statement From Classical Mechanics, Gregory, in the chapter on Hamilton's equations of motion: 14.13: Decide if the energy surfaces in phase space are bounded for the following cases: i.) The two-body gravitation problem with E<0 ii.) The two-body gravitation problem viewed from the...
  43. S

    A Essentially bounded functions and simple functions

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  44. Poetria

    Region bounded by a line and a parabola (polar coordinates)

    Homework Statement ##r=\frac 1 {cos(\theta)+1}## y=-x A region bounded by this curve and parabola is to be found. 2. The attempt at a solution I have found the points of intersection but I am not sure what to do with the line (I need polar coordinates and it is not dependent on r :( )...
  45. S

    MHB On the spectral radius of bounded linear operators

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  46. karush

    MHB 213.15.4.17 triple integral of bounded by cone and sphere

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  47. SemM

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  48. C

    MHB Show that a sequence is bounded, monotone, using The Convergence Theorem

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