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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3
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3answers
97 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
votes
1answer
19 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 ...
0
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0answers
6 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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votes
0answers
20 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 ...
2
votes
0answers
24 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
votes
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 $T$ ...
0
votes
1answer
8 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 ...
-5
votes
0answers
11 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?
2
votes
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 \\ ...
1
vote
1answer
14 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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0answers
31 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 ...
11
votes
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
votes
1answer
10 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. ...
4
votes
2answers
44 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 ...
1
vote
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
votes
1answer
51 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 ...
1
vote
1answer
33 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: ...
1
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0answers
15 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 ...
2
votes
0answers
41 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 ...
1
vote
0answers
16 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.
0
votes
0answers
27 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 $Au \in H^1(0,\infty)$. ...
0
votes
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 ...
0
votes
0answers
16 views

Compact Operators: 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 ...
4
votes
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
votes
1answer
719 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
votes
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
votes
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
38 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 ...
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vote
0answers
21 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 ...
0
votes
1answer
20 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 ...
0
votes
2answers
75 views

Proving a subset of $l_2$ is closed

Let $l_2$ be the set of all real sequences $x=(x_n)$ such that $\sum|{x_n}|^2 <\infty$ and define the norm $||x_n||_2=(\sum\limits_{n=1}^{\infty}|x_n|)^{\frac{1}{2}}$. I want to show that $A=\{ ...
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votes
0answers
11 views

Gateaux but not Frechet differentiable functional [duplicate]

For functional between Banach spaces X,Y: By Gateaux differentiable at $u\in X$ I mean that there exists bounded linear operator $dF(u)$ s.t. $F(u+t\xi)-F(u)=dF(u)\xi+o(t)$ for all $\xi\in X$. For ...
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vote
0answers
59 views
+50

Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
1
vote
1answer
31 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 ...
0
votes
1answer
14 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 $\delta \in (0, \pi)$ ...
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vote
0answers
33 views

Prove that any function f in $L^p$ is the limit, in the metric of $L^p$, of a sequence of simple functions.

I know that I'll need to use dominated convergence here. In the problem, they ask to consider, when f is bounded and nonnegative, the sequence: $s_n(x) = \begin{cases} \frac{i-1}{2^n} \text{ for } ...
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 ...
2
votes
1answer
26 views

$T$ is bijective and homeomorphism.

Suppose $X$ be the set of all polynomial with real coefficients in one variable with norm $$\|p(x)\|=|a_0|+|a_1|+\dots+|a_n|$$ where $p(x)=a_0+a_1x+\dots+a_nx^n$ which induces a metric ...
1
vote
1answer
16 views

which of the following sequences $\{f_n\}\in C[0,1]$ must contain a uniformly convergent subsequence?

Could anyone tell me which of the following sequences $\{f_n\}\in C[0,1]$ must contain a uniformly convergent subsequence? $|f_n(t)|\le 3\forall t\in [0,1],\forall n$ $f_n\in C^1[0,1],|f_n(t)|\le ...
0
votes
0answers
20 views

Is this a Cauchy sequence?

Let $Y$ and $Z$ be Banach spaces. Define $|u|_X := |Au|_Y + |Bu|_Z$ where $A$ and $B$ are linear maps. Suppose I have a sequence $(u_n)$ such that $|Au_n|_Y \to 0$ and $|Bu_n - Bu_m|_Z \to 0$. Does ...
0
votes
0answers
14 views

How to prove this sequence is in $l^2$? [duplicate]

I ran into such a problem in some exercise book on hilbert space. Suppose we have a sequence $ \{a_n \}_1^\infty$. Now, for any sequence $\{ b_n \}_1^\infty $ in $l^2$, we have $$ ...
1
vote
1answer
24 views

On the greatest norm element of weakly compact set

Let $X$ be a Banach space and $K\subset X$ be a nonempty weakly compact set. I would like to know if there exists a point $u_0\in K$ such that $\|u_0\|\geq \|u\|$ for all $u\in K$. Thank you for all ...
0
votes
1answer
12 views

Short proof of sequential Banach Alaoglu for Hilbert spaces

Do you know of a short proof of the fact that bounded sequences in Hilbert spaces admit weakly converging subsequences? If the space is separable, then the common sequential-version proof is what I ...
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votes
1answer
57 views

Approximation Property: Hilbert Spaces [on hold]

Note: This thread is not to gain reputation!! Given a Hilbert space. How to prove: It has the approximation property!
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0answers
14 views

problem about equivalent norms. [duplicate]

Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be equivalent norms on a normed field. Then (i) $\|x\|_1<1$ iff $\|x\|_2<1$; $\|x\|_1>1$ iff $\|x\|_2>1;$ (ii) $\|x\|=1$ iff $\|x\|_2=1$. I want to ...
1
vote
1answer
23 views

Embedding $L^2[0,1]$ into any Hilbert space?

Is it true that every Hilbert space has a closed subspace isometrically isomorphic to $L^2[0,1]$? Can someone sketch a proof of this, or at least point me in the right direction to understanding it? ...
2
votes
2answers
57 views

Fock Space: NESS

Given the CAR-algebra with Hamiltonian dynamics: $$\tau^t[a^\#(\eta)]=a^\#(e^{itH}\eta)\quad(H:\mathcal{D}\to\mathcal{H})$$ (Caution that the Hamiltonian is usually unbounded.) Consider a KMS-state: ...
0
votes
2answers
85 views

Spectral Measures: Riemann-Lebesgue

Problem Given a Hilbert space the Lebesgue measure. Consider a selfadjoint Hamiltonian: $$H:\mathcal{D}\to\mathcal{H}$$ Denote its associated Borel spectral measure by: ...
21
votes
2answers
7k views

Limit of $L^p$ norm

Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$? I ...
0
votes
1answer
28 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 ...