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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Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
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18 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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0answers
8 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

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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2answers
78 views

prove the existence of a measure $\mu$

Suppose $X$ and $Y$ are compact metric spaces and $F : X \rightarrow Y$ is a continuous map from $X$ onto $Y$. If $\nu$ is a finite measure on the Borel sets of $Y$, prove that there exists a measure ...
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70 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
58 views
+50

What is the norm on the completion of a Hilbert space?

Let $X$ be a Hilbert space with a norm $|u|_X = |u|_{X_1} + |Gu|_{X_1}$, where $G:X \to X_1$ is linear and continuous, $X_1$ is a Hilbert space. Define $$|u|_Y = |Gu|_{X_1} + |Tu|_{Z}\quad\text{for ...
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13 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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2answers
38 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
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1answer
22 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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1answer
19 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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1answer
10 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
8 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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0answers
20 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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0answers
29 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 ...
6
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3answers
102 views
+100

Series convergence and compact space

Let $K$ be a compact topological Hausdorff space. $\{x_n\}_1^\infty \subset K $ such that $x_i \not= x_j, i \not=j$ and $\{a_i\}_1^\infty \subset \mathbb{K}$. Show the folowing are equivalent: for ...
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 ...
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0answers
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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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 ...
2
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0answers
184 views

Solve integral(convolution) equation

I have been trying to find a solution to the following convolution equation: \begin{align*} e^{-ax^2/2}*\ln \left(f(x)*e^{-ax^2/2}\right)+e^{-bx^2/2}*\ln \left(f(x)*e^{-bx^2/2}\right)=cx^2 ...
2
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1answer
44 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
23 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
15 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 ...
3
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3answers
101 views

book recommendation on functional analysis

I recently started studying functional analysis. I have many ebooks loaded on my laptop, but can't figure out which one to start with. I've asked my instructor, and he says there aren't any specific ...
3
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1answer
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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0answers
10 views

variational analysis

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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26 views

Approximation Property: Characterization

Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_\varepsilon-1\|_C<\varepsilon\quad(T_\varepsilon\in\mathcal{F}(E))$$ Then every compact ...
6
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1answer
89 views

A problem regarding Functional calulus.

I am stuck in a problem regarding functional calculus.Can anyone help me? Here is the problem. Let $A$ be a Banach Space and $T$ be a bounded operator on $A$. Given that, $\sigma[T]-$ spectrum of $T$ ...
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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
47 views

Different definitions of Morrey and Campanato Spaces

The book by Giaquinta defines Campanato spaces using the seminorm: $$[u]_{p,\lambda} = \left(\sup_{\substack{{x_0\in\Omega \\ ...
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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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38 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
579 views

Hille Yosida theorem application

Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$ u_t + \Delta^2 u = 0 $$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
0
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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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2answers
54 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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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?
2
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1answer
55 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 ...
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1answer
35 views

Partial Isometries: Characterization

Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: $$W^*W,WW^*$$ One derives a partial isometry: ...
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$\ell_q$ 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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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 ...
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0answers
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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0answers
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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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 ...
4
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1answer
1k views

Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
16
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1answer
722 views

Is there a constructive proof of this characterization of $\ell^2$?

I would like to revisit this question, which can be equivalently stated as: Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) ...
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)= ...
2
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0answers
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 ...
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 ...
1
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0answers
22 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 ...