Questions tagged [analysis]
Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).
40,392
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Total differential of the inversion around a circle
I want to compute the total differential of the inversion around the unit circle, i.e.
$$ f: \mathbb{R}^n - \{0\} \to \mathbb{R}^n - \{0\},\quad x\mapsto \frac{x}{||x||^2} = \frac{x}{\langle x, x \...
-1
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13
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Quantitative analysis Statistical test of Likert Scale to determine best option from multiple options
What is the best statistical test to be done to determine the best drawing of three drawings?
Given data collected from a Likert scale on various characteristics of the drawings. Questions on ...
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25
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Using comparison test to tell whether the improper integral converge or diverge
Determine whether the following integral converges or diverges
$\int_{0}^{\pi/2}\frac{1}{sin^2(x)cos^2(x)}$
And assuming I have already proved the comparison tests, and right now I can use comparison ...
- 31
1
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0
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17
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Convexity of balls (induced by a metric) in $C(\mathbb{R})$
I must solve this problem from Rudin's Functional Analysis book. Here $C(\mathbb{R})$ is the space of continuous complex-valued functions with real domain.
I've tried giving many examples of functions ...
- 25
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Is $f\in L^1$ approximated (in the sense of $L^1$) by a sequence of continuous functions?
For arbitrary $f\in L^1$, is there exist a sequence of continuous functions $\{f_n\}_{n=1}^\infty$ s.t. $\lim_n \|f-f_n\|_1=0$ ?
I think this statement is true.
Let $f\in L^1.$
Since $C_c$ is dense in ...
- 411
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0
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8
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How to arbitrarily shift a stock price and an indicator to put them on the same comparison gauge
What to work with
A stock price. Say for example, the S&P 500 Index (SPX) from 2018 to 2019. This gives us some number in the thousands with some nice sideways chop.
An indicator. Say for ...
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1
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27
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A question related to the complex derivative of a function
I am working on the following problem.
Consider a continuous function $\phi:[-1,1]\to \mathbb{C}$ on
$[-1,1]$. Let $$g(z):=\int_{-1}^{1}\frac{\phi(t)}{t-z}\:dt.$$
I am interested in finding the ...
- 418
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16
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A question related to an inequality involving two entire functions
I am currently trying to solve the following problem from a previous qualifying exam.
Let $\alpha,\beta: \mathbb{C}\to \mathbb{C}$ be two non-constant
entire functions with exactly the same zeroes of ...
- 418
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27
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$(\beta_n)_{n \geq 0}$ converges uniformly to the solution of $x' = F(t,x)$, variation of Picard iteration?
This is exercise 2.7. from Differential Equations: A Dynamical Systems Approach to Theory and Practice by Marcelo Viana and José Espinar.
Let $F \colon \mathcal{U} \to \mathbb{R}^n$ be continuous and ...
- 1,066
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2
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23
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Rudin's RCA Fourier Coefficients of $L^1$ - functions
we associate to every $f$ $\in$ $L^1(T)$ a function $\hat f$ on $Z$ defined by
$\hat f $ $=$ $\frac {1}{2\pi}$ $\int_{-\pi}^{\pi}$ $f(t)$$e^{-int} dt$ $(n \in Z)$.
It is easy to prove that $\hat f $ $...
- 751
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Proof of $A\psi = \lambda \psi \Rightarrow h(A)\psi = h(\lambda)\psi$ on Borel functional calculus
Let $A: D(A) \to \mathscr{H}$ be a densely defined unbounded self-adjoint operator on a separable Hilbert space $\mathscr{H}$. By the Spectral Theorem, there exists some finite measure space $(M,\mu)$,...
- 1,601
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14
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Parity of Prolate Spheroidal Wave Functions.
I am trying to find a proof of the parity of the Prolate Spheroidal Wave functions $(\psi_{n})_{n}$, defined as the eigenfunctions of the differential operator
$L_{c}\colon L^{2}[-1,1]\to L^{2}[-1,1]$,...
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50
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Why doesn't Cantor's diagonal argument also work for rational numbers? [duplicate]
Let's suppose that we don't know how to arrange all the rational numbers in such a way that easily shows that they're countably infinite. I present to you the following list which claims to include ...
- 117
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21
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LCH space and radon measure
I'm trying to solve this exercise, but I have some doubts: I think point (b) is correct, but I'm not sure about point (a).
Let X be a locally compact Hausdorff topological space and let µ: S → [0, +∞] ...
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0
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34
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Duality of $H^1$ and BMO.
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
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45
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$\mathbb{C}$ is complete
Can
$\mathbb{R}^2 \simeq \mathbb{C}$ and because $\mathbb{R}^2$ is complete $\implies \mathbb{C} $ is complete
be an argument to show thaf $\mathbb{C}$ is complete? Or can you give me a proof please?...
- 397
1
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1
answer
97
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I derived a formula for $[x!]^\prime$. Is it correct?
The starting point was that $ \Gamma'(x+1)=\Gamma(x+1)\psi(x+1)$ where $\psi(x+1)=-\gamma+H_{x}$ . Hence $$ [x!]' = x!\biggl[-\gamma+\sum_{k=1}^{x}\frac{1}{k}\biggl]$$ For example $ [4!]' = 24[-\gamma+...
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30
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Condition $\sum_{n=1}^\infty \left[ \sum_{m=1}^\infty a_{m,n}\right]$ and $\sum_{m=1}^\infty \left[ \sum_{n=1}^\infty a_{m,n}\right]$ to be equal
Consider real numbers $a_{m, n}$ for $n = 1 ,2,...$ and $m = 1, 2,...,$ an assume that inner and outer sums in the expressions
\begin{align}
A:= \sum_{n=1}^\infty \left[ \sum_{m=1}^\infty a_{m,n}\...
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Strassmann's thoerem and irrationality measure of certain number
In this note from Keith Conrad, he explains an interesting application of Strassmann's theorem to the divergence of certain linear recurrence integer sequence. More precisely, the sequence defined as $...
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2
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27
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How can I calculate the dual norm in this case?
We define $E$ the space of continuous functions as $x: [0,1] \rightarrow \mathbb{R}$, endowed with the norm for $x \in E$:
$$\|x\|_{\infty} = \max_{t\in [0,1]}|x(t)|$$
It is further defined $T : E\...
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1
answer
13
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Help with determining constant values for water depth measurements using trigonometric functions
I am currently working on a problem related to water depth measurements and I am seeking some help with it. The problem is as follows:
Water depth at a dock is measured during the first 16 hours of a ...
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1
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0
answers
24
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Lipschitz continuity of translation in $L^p$
Let $f\in L^p(\mathbb{R})$ for $1\leq p <\infty,$ then $||f(x+h)-f(x)||_{L^p(\mathbb{R})} \rightarrow 0$ as $h\rightarrow 0$.
Furthermore, for $p=1,$ if $u\in L^1(\mathbb{R}) \cap W^{1,1}(\mathbb{R}...
- 364
1
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0
answers
25
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Using Mass distribution principle to provide lower bound for the Hausdorff dimension of Cantor set
given the (ternary) Cantor set $\mathcal{C}$ it is well known that its Hausdorff dimension is given by $\dim_\mathcal{H}(\mathcal{C})=ln(2)/ln(3)$, which I am going to denote by $\alpha$. I am ...
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18
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Principle value vs Lebsgue integrability
Though a large class of functions are Lebesgue integrable, for certain class of functions, say for example $f(x)=1/x$ Lebesgue integral over the interval $[-a,a]$ for $a>0$ is undefined. But the ...
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On a formula for Hypergeometric function $_3F_2(a,b,c;d,e;1)$
I need to prove a formula for the Hypergeometric function $_3F_2(a,b,c;d,e;1)$
I searched and got the following from Wolfram alpha ( see:- second formula in https://functions.wolfram.com/...
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58
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Another proof of e is irrational?
I find some proof here, but can I prove it in the following way?
Assume $e$ is rational, then $e=\frac{p}q$, both $p$ and $q$ are positive integers. By Lagrange's Remainder Theorem,
$$e^x=1+x+\frac{1}{...
- 7,007
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1
answer
96
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Uniform convergence of $\sum_{n=1}^\infty x \sin\frac{1}{x^2n^2}$, $x \in (0,+\infty)$
Consider a series $\sum_{n=1}^\infty f_n(x)$, where $f_n(x) = x \sin\frac{1}{x^2n^2}$ and $x \in (0, +\infty)$. If one fix an arbitrary $x \in (0, +\infty)$, then for sufficiently large $n \in \mathbb{...
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$f(m+n)+f(mn)=f(m)f(n)+1$ [closed]
How to find all functions $f:R\rightarrow R$ such that
$f(m+n)+f(mn)=f(m)f(n)+1$
I I've tried a lot, but I didn't find a solution
I only know that $f(x)=1 $ and $f(n)=n+1$ are solutions
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0
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45
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Prove $1+ \sum_{m=1}^{n} \frac{1}{2^{m-1}} < 3$ [closed]
I would like to prove the following:
$$
1+ \sum_{m=1}^{n} \frac{1}{2^{m-1}} < 3
$$
So I know the series is a geometric series. However I am getting that it sums to 2. Which cannot be true since I ...
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0
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61
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What is the symbol for a function that is complex differentiable?
Suppose $f:D \subset \mathbb{R}^2 \to \mathbb{R}^2$.
If $f$ is differentiable (or $\mathbb{R}$-differentiable) we say $f \in C^1(D;\mathbb{R}^2)$ or $f \in C^1(D)$ (meaning the partial derivatives ...
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1
answer
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Does the shadowing property hold for the rotation?
Let $\alpha\in \Bbb{R}$ and consider the rotation $R_\alpha: \Bbb{R}/\Bbb{Z}\rightarrow \Bbb{R}/\Bbb{Z}$ a.t. $x\mapsto (x+\alpha)\operatorname{mod}\Bbb{Z}$. I want to check if $R_\alpha$ satisfies ...
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1
answer
46
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$B(0,1) $ isn't totally bounded in $\ell^{1}$ [closed]
I've been able to show using Riesz's lemma that $B[0,1]$ isn't totally bounded , and I tried to solve it for $B(0,1)$ , but I wasn't capable of solving it.
Any suggestions?
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A differential equation with infinitely many characteristic roots
Let us consider the following neutral differential equation :
$$ \dot{x}(t)-\dot{x}(t-\tau)=ax(t)+bx(t-\tau)$$
where $\tau >0$ is a constant delay, and $a,b\in \mathbb{R}$.
It is a well-known fact ...
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1
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31
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Rudin’s RCA A Convergence Problem Fourier Series
There is it: Is it true for every $f\in C(T)$ that the Fourier series of $f$ converges to $f(x)$ at every point $x$?
Let us recall that the $n$th partial sum of the Fourier series of $f$ at the point $...
- 751
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1
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43
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how to prove any straight line can divide trapezoid be two equal area?
Intermediate value theorem quarantee that there is infinite line can divide trapezoid to be two equal area. I confused the properties and how to generalization (without picture) ensure that the line ...
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0
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$ \frac{1}{t_n} \int_0^{t_n} e^{2 \pi i h f(x)} \, dx \to 0$ if $ \frac{1}{n} \int_0^n e^{2 \pi i h f(x)} \, dx \to 0 $
Suppose that $f$ is a measurable functions on the real line and $h \in \mathbb Z \setminus \{ 0 \}$ such that
$$
\frac{1}{n} \int_0^n e^{2 \pi i h f(x)} \, dx \to 0
$$
as $n \in \mathbb N$ tends to ...
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0
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52
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How to fill in this proof of Jensen's inequality for conditional expectation?
Recall Jensen's inequality for conditional expectation: $X$ an integrable (or nonnegative), real-valued, $\mathfrak{F}$-measurable random variable. $\varphi: \mathbb{R} \to \mathbb{R}$ convex. $ \...
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1
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50
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Can we decompose complex differential equation into two real?
there is a simple system:
$$
-y''(x) = f(x)
\\
y(0) = 0
\\
y(1) = 0
$$
where $y$ and $f$ is complex function of real variable $x$.
Is it legal to decompose this functions into two real (four real for ...
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votes
1
answer
46
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Existence of "induced measure" on fibers of a measurable function between measure spaces?
Let $f : X\rightarrow Y$ be a measurable function between measure spaces $X,Y$ with measures denoted $\mu,\nu$ respectively. Suppose singleton subsets of $Y$ are measurable; hence fibers of $f$ are ...
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Construction of differentiable function $f$ such that $g(x)=-x\ln f(x)$ is convex.
I am following a proof of the characterization of uniformly integrable functions by test functions, and the proof first proves the following lemma.
Given $f_0:[0,+\infty)\to[0,+\infty)$ to be non ...
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0
answers
24
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Newton's method at log scale
For a set of system of equations
$$f_i(\vec{x})=0$$
The newton's method was
$$\vec{x}_{n+1}=\vec{x}_{n} -(J_{f_i}(\vec x_n)) \vec{f}_i(\vec x_n)$$
where $J_{f_i}(\vec x_n)$ was the Jacobian of $\vec{f}...
0
votes
0
answers
18
views
Properties of $a(t) = g(\log(t+1))$ with $g : [0,\infty) \to \mathbb R$ a strictly increasing and convex [closed]
Consider the function $a(t) = g(\log(t+1))$ with $g : [0,\infty) \to \mathbb R$ a strictly increasing convex functions. It is well-known that under the above assumptions the derivative of $a$ exists ...
- 67
1
vote
1
answer
139
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Product Law of Exponents holds for negative base? [closed]
For all $b>0$ and $x,y \in \mathbb R$, $$b^xb^y=b^{x+y}.$$
Does the above also hold for $b<0$? (If yes, give a proof. If no, give a counterexample.)
- 392
1
vote
2
answers
90
views
Two identities of sums and integrals
I'd like to know how to show the following identities.
$$ \sum_{n\ge 0}\frac{(-1)^{n+1}}{(4n+1)^2}=\int_{0}^{1}\frac{\log x}{1+x^4}dx, $$ and
$$ \sum_{n\ge 0}\frac{(-1)^{n+1}}{(4n+3)^2}=\int_{0}^{1}\...
- 181
0
votes
0
answers
42
views
Is it true that If $A + A = 2A$ in a TVS then $A$ is convex? what if we assume A is complete?
We know that if $A$ is convex then it implies that $ A + A = 2A$ by simple arguments.
$2A \subset A + A $ this always holds.
Now
Let $ u+v \in A + A $ for some $u,v \in A$
Then $ (u+v)/2 \in A $ since ...
0
votes
1
answer
17
views
$f(x, y, z) := xy + z^2$ on unit circle $B := \{(x, y, z) \in \mathbb{R}^3, x^2 + y^2 + z^2 \leq 1\}$
We look at $f(x, y, z) := xy + z^2$ on unit circle $B := \{(x, y, z) \in \mathbb{R}^3, x^2 + y^2 + z^2 \leq 1\}$.
A) Explain why $f$ has a minimum and maximum on $B$.
B) Show that $f$ has no local ...
- 153
2
votes
2
answers
64
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Calculate $\iint_R \cos\left(\frac{y}{x}\right)\text{d}A$
Hello I would like to know if I answered the question correctly,
Calculate $\iint_R \cos\left(\frac{y}{x}\right)\text{d}A$ with $R$ the region under the graph of $y = x^3$, between $x = 0$ and $x = 1$...
- 153
-3
votes
0
answers
29
views
DDPM loss function KL divergence [closed]
Given this DDPM loss function:
$$
\begin{aligned}
&\mathbb{E}_q\left[\log{\frac{q\left(x_T\mid x_0\right)}{p_\theta\left(x_T\right)}} + \sum_{t=2}^T \log{\frac{q\left(x_{t-1}\mid x_t, x_0\right)}{...
- 1
0
votes
0
answers
27
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Estimate $\int_a^b \frac{[-t^{\frac{8}{3}}+t^{\frac{2}{3}}T^2-1]^{\frac{3}{2}}}{t^5}dt$ as $T\to \infty$ where $a,b$ are two roots of the numerator?
For $t,T>0$, let $f(t):=-t^{\frac{8}{3}}+t^{\frac{2}{3}}T^2-1$. By analyzing the derivative $f'(t)=-\frac{8}{3}t^{\frac{5}{3}}+\frac{2}{3}t^{-\frac{1}{3}}T^2$, we see on $(0,\infty)$, $f(t)$ ...
- 359
0
votes
0
answers
50
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Rudin's RCA $5.10$ theorem
This is the definition which we need for the theorem:
This is the theorem $5.9$:
There is $5.10$:
If $X$ and $Y$ are Banach spaces and if $\Lambda$ is a bounded linear transformation of $X$ onto $Y$ ...
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