Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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13 views

limit of a function when x goes to infinity

Let $f\colon [0,\infty)\to\mathbb{R}$ be differentiable on $(0,\infty)$, and assume that $f^\prime(x)\xrightarrow[x\to\infty]{}b$. show that for any $h>0$ ,we have $\frac{f(x+h)−f(x))}{h} ...
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13 views

Properties of sup and lim inf.

Let $(a_{n})_{n \geq1}$ be a sequence of numbers such that $a_n\leq M$ for all $n \geq 1$ . Prove that $$ \lim_{n\to\infty} \inf \{a_n,a_{n+1},...\} = \sup_{n \geq1} \inf\{a_n,a_{n+1},...\} $$ ...
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16 views

Isomorphisms with invariant linearly independent dense subset.

If $T$ is an isomorphism acting on a separable Banach space $X$, can we find a countable, dense, linearly independent set $D\subset X$ such that $T(D)=D$? If $X$ is finite dimensional, then the answer ...
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25 views

about Heine-Borel Theorem in a function space

In Pugh's real mathematical analysis. About the Heine-Borel Theorem in a function space, it states that a subset $\epsilon$ $\in C^0$ is compact if and only if it is closed, bounded, and ...
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27 views

Every isomorphism on a separable Banach space has a completely invariant dense subset

If $T$ is an isomorphism acting on a separable Banach space, can we always find a countable dense subset $D$ of $X$ such that $T(D)=D? $
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11 views

Let $B = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2| \}$. Show that B is balanced, but that its interior is not.

I have the following definitions. The interior $E^o$ of $E$ is the union of all open sets that are subsets of $E$. A set $B \subset X$ is said to be balanced if $\alpha B \subset B$ for every ...
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0answers
8 views

Conditional convergence of series and product

Let $e_k = 0$ for $k$ is odd and $e_k = 1$ when $k$ is even. Set $b_k = \frac{e_k}{k} + \frac{(-1)^k}{\sqrt{k}}$. How do I show that the series $\sum b_k$ diverge while the corresponding product $\Pi ...
5
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1answer
53 views

References for hemicontinuity?

Let $X$ be a real vector space, $K\subset X$ be a nonempty and convex set. The mapping $f:X\rightarrow\mathbb{R}$ is said to be hemicontinuous if for every $u,v\in K$, the mapping ...
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0answers
29 views

Orthogonality in Space of Polynomials of Degree at Most 2

Let $E$ be the space of polynomials of degree at most $2$. On $E$ define $\langle f,g \rangle := f(-1)\overline{g(-1)}+f(0)\overline{g(0)}+f(1)\overline{g(1)}$ for $f,g \in E$. a). Show that this ...
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15 views

Functional Analysis/ Toplogy

The Mackey Arens theorems gives us the existence of the Mackey topology $\tau(L^{\infty},L^1)$ which is the strongest locally convex topology we can put on $L^{\infty}$ in order to make $L^1$ the dual ...
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70 views

Integration by parts for weak derivatives

I'm trying to show that if $g$ is such that $f(b) - f(a) = \int_a^b g(t) dt$ for any $a<b \in \mathbb{R}$ then we have: (for $f, g \in L^2(\mathbb{R})$) $$\int_a^b f(t)g(t) = \frac{1}{2}(f(b)^2 - ...
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0answers
12 views

Comparison of pseudomonotone definitions

Are the intersections $\bigg(\cdot\bigg) \bigcap[x,y]$ necessary for the terms on page 3? Or could the proof follow by dropping the $[x,y]$? See paper. Thanks
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1answer
20 views

Proof of Gelfand formula for spectral radius

STATEMENT: Let $A$ be a Banach algebra, then for every $x\in A$ we have $$\lim_{n\rightarrow\infty}||x^n||^{1/n}=r(x)$$ Proof: We know that $r(x)\leq \lim \inf_n||x^n||^{1/n}$, so it suffices to ...
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1answer
23 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial ...
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0answers
9 views

Shift operators and c0 semigroups

I am asked where the bounded continuous functions on $\mathbb{R}$ (with sup norm) with the right shift operators ( $T_t(f)(x)=f(x-t)$ ) form a c0 semigroup. I believe the answer is no, but I am ...
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1answer
12 views

Closed linear operator

I am having some trouble in showing the following map is closed: For $f\in L^2(\mathbb{R^2})$ with $(x+iy)f(x,y)\in L^2(\mathbb{R^2})$, $M(f)(x,y)=(x+iy)f(x,y)$. I am also asked to find the ...
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0answers
24 views

Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
2
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1answer
19 views

Is $X= \{ u \in H^1(\Omega \times I) \mid \int_\Omega u(x,y)\;\mathrm{d}x = 0 \text{ for a.a. $y \in I$}\}$ a Hilbert space?

Let $\Omega$ be a bounded domain and let $I$ be an unbounded interval. Let $$X= \{ u \in H^1(\Omega \times I) \mid \int_\Omega u(x,y)\;\mathrm{d}x = 0 \text{ for a.a. $y \in I$}\}$$ Is this ...
2
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0answers
30 views

Separating the integral of a product of functions apart

Show that $$\left(\frac{1}{\pi}\int_{-\pi}^{\pi}(f(x+t))^2(K_n(t))^2\mathop{dt}\right)^{1/2}\leq \left(\frac{1}{\pi}\int_{-\pi}^{\pi}(f(x+t))^2\mathop{dt}\right)^{1/2}$$ where $K_n(t)$ is Fejer's ...
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2answers
20 views

Is distributional Derivative of $\delta^{(2)}(-x)=-\delta^{(2)}(x)$

Is distributional Derivative of $\delta^{(2)}(-x)=-\delta^{(2)}(x)$ ?? or $\delta^{(2)}(-x)=\delta^{(2)}(x)$ ?? I know that $\delta(-x)= \delta(x)$ and $\delta^{(1)}(-x)=-\delta^{(1)}(x)$. How ...
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0answers
20 views

Mountain pass theorem

Let $I$ be a real functional over a Hilbert space $H$, satisfying all the conditions in the Mountain pass (M-P) theorem. My question is, can the assumption in the M-P theorem that $I[v]\leq 0$ for a ...
3
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86 views

Question about computing a Complicated integral

where $\beta$ is defined like this: I'm trying to prove (2.18) but i don't know how to do, i calculated the integral but i don't find anything %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EDIT1: ...
2
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1answer
46 views

How to find the norm of $ \Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$ in $\ell^2$

Suppose that $(x_n)$ is a sequence in $\ell^2$, i.e. $\displaystyle \sum_{i=1}^{\infty} x_n^2 < \infty$. Define: $$\Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$$ Find $\| \Lambda \|_2$ ...
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1answer
27 views

What is the completed projective tensor product of compactly supported smooth functions on two manifolds?

Let $M$ and $N$ be smooth manifolds (not necessarily closed). It is a lovely fact that $$C^\infty(M \times N) \cong C^\infty(M) \hat{\otimes}_\pi C^\infty(N).$$ See, for the instance, the book ...
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0answers
16 views

About Lusternik-Schnirelmann category

I' studying this paper: http://www.sciencedirect.com/science/article/pii/S0022039608003744 In page 1303-1304 they defined two functions $\phi_{\varepsilon}$ and $\beta$ But i don't understand ...
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12 views

variational analysis [on hold]

Let $I$ be a functional over a Hilbert space, as in the Mountain pass theorem. Can the condition that there exists $v$such that $I(v)\leq 0$ for $||v||>r$ be replaced by $I(v)=0$?.
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28 views

Dense subset in sequence space

I'm trying to prove that $F=\{x=\{x_n\}_{n\in \mathbb{N}}\in l^2(\mathbb{N}):\sum_{n=1}^{\infty} x_n=0\}$ is dense in the sequence space $l^2(\mathbb{N})$. I think it should be an easy exercise, but ...
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31 views

Mackey Topology

Let $C$ be a convex subset of the unit ball of $L^{\infty}$. Show that if $C$ is closed in the topology induced by the standard $\|\cdot\|_p$ norm for some $p>1$, then $C$ is closed in the Mackey ...
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1answer
24 views

Borderline case of interpolation of Banach spaces

Let $B \subset A$ be Banach spaces with a continuous embedding. Is the inequality $$ \|b\|_B \leq C \sup_{t > 0} \inf_{\tilde{b} \in B} \{ \|b - \tilde{b}\|_B + t \|\tilde{b}\|_A \} \quad \forall b ...
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15 views

Automotive accident question [on hold]

A car traveling 14 mpg that weighs 2,405lbs strikes another car that has it's brakes applied the weighs of the second car is 4,170lbs. How far will the second travel after impact?
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1answer
16 views

Is complemented subspace of complemented subspace is complemented? [on hold]

Let $X\subset Y\subset Z$ be Banach spaces such that $X$ is complemented in $Y$ and $Y$ is complemented in $Z$. Is it true that $X$ is complemented in $Z$?
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44 views

Derivations: Characterization

Given a smooth manifold. (In fact, it seems irrelevant to regard manifolds.) Regard germs of functions: $$\mathcal{C}_p^\infty(M):\quad f\sim g:\iff f\restriction\equiv g\restriction$$ and the ...
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1answer
15 views

Is $H^1(0,\infty) \subset C^0([0,\infty))$?

Is it true that $H^1(0,\infty) \subset C^0([0,\infty))$ is a continuous embedding? How would I prove it? I do know this holds for bounded domains in one dimension but here we have the half line. ...
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0answers
17 views

Functional dual question

Is it true that $L^1$ is the dual of $(L^{\infty}(\Omega),||\cdot||_p)$ for any $p > 0$, where the p-norm symbol denotes quasinorm for $p <1$, where $\Omega$ has finite measure?
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18 views

Prove that $\left\{e^{i n t}\right\}_{n\in\mathbb Z}$ is a Riesz basis on $L^2[-\pi,\pi]$.

Prove that $$\left\{e^{i n t}\right\}_{n\in\mathbb Z}$$ is a Riesz basis on $L^2[-\pi,\pi]$. Can I have any reference or any suggepstion please? Thanks.
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30 views

Why is this integral involving the mean value function zero?

Let $u$ and $v$ belong to $H^1(\Omega \times (0,\infty))$ on a bounded domain $\Omega$. Define $$(Au)(y) := \frac{1}{|\Omega|}\int_\Omega u(x,y)\;\mathrm{d}x.$$ We have that $Au \in H^1(0,\infty)$. ...
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0answers
18 views

$\ell_q$ is not finitely representable in $\ell_p$ if $2<q<p$.

This seems to be a well known result in Banach space theory. It is referenced, for example, in Pietsch's book "History of Banach spaces and Linear Operator". Where can I find a proof? Who was the ...
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0answers
20 views

Approximation Property: Decomposition

This is a real question of me. Given a Banach space. Consider a basis on finite dimensional range: $$\dim\mathcal{R}F<\infty:\quad y_1,\ldots, y_N$$ Hahn-Banach lifts the dual basis up: $$ y_n\in ...
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2answers
55 views

Why are functional analysts interested in not only the point spectrum of $f$, but also, its spectrum?

Suppose $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\},$ that $X$ is a Banach space over $\mathbb{K}$, and that $f : X \leftarrow X$ is a bounded linear transform. Then the spectrum of $f$ is defined as the ...
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2answers
28 views

Understanding part of a theorem of Calculus of Variations

I have trouble understanding the following statement (From Gelfland's Calculus of Variations book): If $\phi[h]$ is a linear functional and if ...
2
votes
1answer
56 views

Proof Involving Generalized Mean

Let $x=(x_1,...,x_n) \in \mathbb R^n$ and $$g(p)=\sqrt[p]{\frac{1}{n}\sum_{k=1}^{n} |x_k|^p)}$$ Using Hölder's inequality, show that $g(p)$ is increasing on $(0,\infty)$. For a sequence with ...
3
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1answer
38 views

Proving a metric on X.

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= ...
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0answers
23 views

For a Banach algebra $\mathcal{A}$ and an idempotent $e$, show that $\mathcal{A}e$ is a Banach algebra

I am studying the spectral theory of operators on Banach and Hilbert spaces, reading through Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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17 views

Is the system $\left\{\frac{e^{i n t}}{2\pi}\right\}_{n\in\mathbb Z}$ a Riesz basis on $L^2(-\pi,\pi)$?

Is the system $$\left\{\frac{e^{i n t}}{2\pi}\right\}_{n\in\mathbb Z}$$ a Riesz basis on $L^2(-\pi,\pi)$? I think not because $$\frac{1}{2\pi}\int_{-\pi}^\pi \frac{e^{i (n-m) t}}{4\pi^2}dt\neq 1$$ if ...
0
votes
1answer
22 views

Why isn't the completion of $C^0$ wrt. the $L^2$ norm a space of sequences instead of a space of functions?

We know that $L^2(\Omega)$ can be defined as the completion of $C^0(\Omega)$ with respect to the norm $$\left(\int_\Omega |u|^2\right)^{\frac 12}.$$ But strictly speaking, $L^2(\Omega)$ is a space of ...
2
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0answers
46 views

Fourier transform of $e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} $

I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of ...
3
votes
1answer
39 views

Verifying a Vector Space Via Given Axioms

Let $X$ be the collection of all sequences $\{\alpha_n\}_{n=1}^{\infty}$ of scalars from $\mathbb{K}$ such that $\alpha_n=0$ for all but a finite number of values of $n$. Define addition and scalar ...
0
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1answer
18 views

Question about a proof of Riemann localization theorem

The Riemann Localization Theorem states that Let $f \in L_{2 \pi}^2$ and $x_0 \in \mathbb R$. Then $$ \lim_{n \to \infty} (S_nf)(x_0) = f(x_0)$$ if and only if there is a $\delta \in (0, \pi)$ ...
1
vote
1answer
24 views

Why is $J(u) := \int_\Omega |\nabla u|^2$ convex?

Define $J(u) := \int_\Omega |\nabla u|^2$ over $\{ u \in H^1(\Omega) : tr(u) = g\}$. Why is $J$ convex? I keep getting $J(tu + (1-t)v) \leq 2t^2J(u) + 2(1-t)^2J(v)$ by using the triangle inequality ...
1
vote
1answer
33 views

Understanding a proof from Conway: showing existence of idempotents using functional calculus

I am studying the spectral theory of operators on Banach and Hilbert spaces, making use of Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...