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.
##\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...
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...
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##.
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...
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...
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...
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...
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...
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...
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?
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...
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}...
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...
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...
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...
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...
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}} ##...
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$.
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...
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...
##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##...
I am trying to solve the following problem. Let us take a bounded domain $S$ in which an explosive device is located. A team is deployed to locate and disable the device before a certain time T (when the device explodes). There are several criteria to be satisfied:
1. The domain $S$ is...
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...
I don't know how to show that this limit is zero.
It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one.
Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.
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...
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...
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...
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...
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...
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...
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##...
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##...
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
$$
X[k] = \sum\limits_{k =...
Homework Statement
Let R be the area in the xy-plane in the 1st quadrant which is bounded by the curves y^2+x^2 = 5, y = 2x and x = 0. (y-axis). Let T be the volume of revolution that appears when R is rotated around the Y axis. Find the volume of T.
Homework EquationsThe Attempt at a Solution...
Sorry for all the questions. Reviewing for my midterm next week. Fun fun.
If someone could take a look at my proof for (a) and help me out with (b) that'd be awesome!
(a) Let $\Delta$ be a partition of $[a, b]$ that is a refinement of partition $\Delta'$. For a real-value function $f$ on $[a...
Define $f(x)=sinx$ on $[0, 2\pi]$. Find two increasing functions h and g for which f = h−g
on $[0, 2\pi]$.
I know that if f is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions. However, we didn't do any examples of this in class. Is there a...
Homework Statement
Let G be the region bounded by the planes x=0,y=0,z=0,x+y=1and z=x+y.
Homework Equations
(a) Find the volume of G by integration.
(b) If the region is a solid of uniform density, use triple integration to find its center of mass.
The Attempt at a Solution
[/B]
My...
Hello! (Wave)
I want to show for the initial value problem of the wave equation
$$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$
that if the data (i.e. the initial data and the non-homogeneous term $f$) have compact support, then, at each time, the solution has also compact support.
I...
Homework Statement
The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong.
Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
Homework Statement
From Classical Mechanics, Gregory, in the chapter on Hamilton's equations of motion:
14.13: Decide if the energy surfaces in phase space are bounded for the following cases:
i.) The two-body gravitation problem with E<0
ii.) The two-body gravitation problem viewed from the...
Homework Statement
##r=\frac 1 {cos(\theta)+1}##
y=-x
A region bounded by this curve and parabola is to be found.
2. The attempt at a solution
I have found the points of intersection but I am not sure what to do with the line (I need polar coordinates and it is not dependent on r :( )...
Hi EVERYBODY:
General knowledge: The homogeneous linear Fredholm integral equation
$\mu\ \varPsi(x)=\int_{a}^{b} \,k(x,s) \varPsi(s) ds$ (1)
has a nontrivial solution if and only if $\mu$ is an eigenvalue of the integral operator $K$. By multiplying (1) by $k(x,s)$ and...
$\textsf{Find the volume of the given solid region bounded by the cone}$
$$\displaystyle z=\sqrt{x^2+y^2}$$
$\textsf{and bounded above by the sphere}$
$$\displaystyle x^2+y^2+z^2=128$$
$\textsf{ using triple integrals}$
\begin{align*}\displaystyle
V&=\iiint\limits_{R}p(x,y,z) \, dV...
Hi, why do unbounded operators and bounded operators differ so much in terms of defining their spectra?
1. The unbounded operator requires a self-adjoint extension to define its spectrum.
2. A bounded one does not require a self-adjoint extension to define the spectral properties.
3. Still the...
Dear Every one,
In my book, Basic Analysis by Jiri Lebel, the exercise states
"show that the sequence $\left\{(n+1)/n\right\}$ is monotone, bounded, and use the monotone convergence theorem to find the limit"
My Work:
The Proof:
Bound
The sequence is bounded by 0.
$\left|{(n+1)/n}\right|...