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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Closest element to a subset of $\mathbb R^2$

Let $U=\{(x,y)|x,y\geq 0\}$ be a closed convex subspace of $(\mathbb R^2,\|\cdot\|_\infty )$. Show that the closest elements in $U$ to $(1,-1)$ are $\{(x,0)|0\leq x\leq 2\}$ Show that the closest ...
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1answer
16 views

A bounded linear functional on a Hilbert space that is a Hahn-Banach extension of one on a subspace

Let $M$ be a closed linear subspace of a Hilbert space $H$ and $g\in M*$(all bounded linear functional on $M$). Let $\pi$ be the orthogonal projection of H onto M, then $f=g\circ\pi$ is the only ...
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34 views

Convergence on $\ell^p$ spaces

I have been working on this problem. Show the following are equivalent: i. $\sup_{n\in\mathbb{N}}\|x_n\|_p<\infty$ and $\lim_{n\to\infty}x_n(j)=0$ for each $j\geq1$, ii. ...
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17 views

Can dimension counting argument generalize to functional space

By dimension counting I mean the following argument: there is no injective continous mapping from any open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$ if $n>m$ (is this true? I can't give rigous ...
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2answers
74 views

Prove for a close subset of $\mathbb R$

Let $S=\{x\in \mathbb{R}\mid 0\leq x\}$. Prove that $S$ is a closed subset of $\mathbb{R}$. I know I need to show that $\forall x \exists x_n \xrightarrow[n \to \infty]{} x\in S$, but I have no ...
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15 views

On bounded linear operators not attaining its norm

Suppose $T$ is a bounded linear operator on a Hilbert space $H$ which does not attain its norm. Does there exist a sequence $\{e_n\}_{n\in\Bbb N}$ of orthonormal vectors such that $\lVert Te_n\rVert ...
2
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1answer
90 views

What does $ f ^ {n} (x ^ {1/n}) = … $ mean?

I was asked to check whether the sequence of functions $ \{ x_{n} (t) \} $ defined as $$ x ^{n} _{n}(t ^ \frac{1}{n}) = \begin{cases}n, & t \leq \frac{1}{n} \\\frac{1}{n},& t > ...
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1answer
25 views

Finding the closest element to a function in a normed space containing functions.

Let $B=\{f\in c[1,0]|\forall 0\leq x\leq 1 : f(x) \geq 0\}$. Given $f\in c[0,1]$, find the closest element in $B$ under the $\|\cdot\|_2$ norm. I can see something similar to this question in the ...
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2answers
27 views

How to prove $r(T_1 + T_2) \leq r(T_1) + r(T_2)$ when $T_1T_2 = T_2T_1$ for bounded linear operators?

Suppose $T_1$ and $T_2$ are bounded linear operators in a complex banach space and $r(A)$ is the spectral radius of $A$, satisfying $$ r(A) = \inf_{n>0} \|A^n\|^{1/n} = \lim_{n\rightarrow \infty} ...
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16 views

Applications of Stein's Interpolation Theorem

Are there any neat applications for Stein's interpolation theorem, especially in the context of complex analysis? There's a lot of proofs online but I couldn't find many "corollaries" that follow from ...
4
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1answer
36 views

What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$?

What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$? I've seen this answer but this is on an infinite domain. I'm interested only in $(0,1)$. I tried playing around with ...
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1answer
11 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
3
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1answer
66 views

Is $C^\omega([0,1])$ normable? (And about the growth of coefficients of infinitely differentiable functions)

This question arised to me when trying to prove that the space of infinitely differentiable functions defined in a compact space $K\subset\mathbb{C}$ taking values in $\mathbb{C}$, that is ...
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1answer
30 views

Hahn Banach theorem proof

Let $X$ be real linear space and $X_{0}$ is its subspace. Also let $p$ be a finite convex functional in $X$ and $f_{0}$ is linear functional in $X_{0}$, while $f_{0}(x)\le p(x),x\in X_{0}$ Then ...
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0answers
15 views

In separable, inner-product space X every complete orthonormal set is closed and vice versa

English is not my native language so if anything needs definition just tell me. I have a problem with proving that if set is closed then it is complete Let $\{\varphi _{k}\}_{k=1}^{\infty }$ be an ...
3
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47 views

proving compactness with arzela-ascoli

Let $C>0$ and $$ M=\left\{f\in C^1[0,1]:\int_0^1|f(x)|^2dx+\int_0^1|f'(x)|^2dx\leq C \right\} $$ Is $\overline M$ compact in $C[0,1]$? I think this follows by the Arzela-Ascoli theorem. How can ...
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3answers
41 views

Definition of a closed subset of $\mathbb R$

Let $A=\{x\in \mathbb R \mid 0\leq x\}$. Prove that $C$ is a closed subset of $\mathbb R$. As far as I understand a closed set/subset is a set that for all sequences within it are bounded and the ...
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2answers
27 views

Bessels inequality

Why can you change the top index of sum in the last step of this proof (see below) from $n$ to infinity ? Let $r_n = x - \sum_{k=1}^{n} \langle x,e_k \rangle \cdot e_k$. Then for $j = ...
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1answer
9 views

Weakly square summable series as operators on Hilbert spaces

Let $H$ be a Hilbert space and let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence in $H$ such that $\sum^{\infty}_{n=1}|\langle h,a_n\rangle|^2<\infty$ for all $h\in H$. Here ...
3
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1answer
145 views

Show that if $[Q,P]=it\Bbb{I}$ then the operators are unbounded

In the Hilbert space $\mathcal{H} = L^2(\mathbb{R},dx)$, let 2 symmetrical operators $P$ and $Q$ be given, with the following properties: $D(P) = D(Q) = \mathcal{S}(\mathbb{R})$ ...
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17 views

What is the discretization matrix of 2D Poisson equation of finite diffence with checkerboard (black and red) pattern?

Given the problem$-\Delta u(x,y)=f(x,y)$ on unit rectangle $\Omega=[0,1]^{2}$ and $u(x,y)=g(x,y)$ on $\partial\Omega$, what is the finite difference matrix associated with step size $h=1/(2N+1)$ where ...
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1answer
26 views

Theorem 2.14 (The dual of $L^p(\Omega)$) in Lieb's Analysis book

The following pictures are Theorem 2.14 (The dual of $L^p(\Omega)$ in Lieb's Analysis book and its proof of the case $1<p<\infty$. My question is how to get the inequility (3) in the red box? ...
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0answers
10 views

Control the value of a function at a point by the norm of its fourier transformation and itself

$n\leq 3$ ,$\Delta$ is the Laplacian on $L^{2}(R^{n})$, $Dom(\Delta) = \{\phi\in L^{2}(R^{n})|\Delta\phi\in L^{2}(R^{n})\}$. Please show that:for any $\phi\in Dom(\Delta)$,there exists constants ...
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0answers
24 views

Proof that addition on a Banach space is continuous

What I have so far: Let $(W,+,\cdot,\Vert\cdot\Vert)$ be a Banach space. We have the map $$+ : W\times W \to W.$$ The topology $T$ on $W$ is given by $$T = \{U\subseteq W ~\big|~ \forall u\in U : ...
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0answers
41 views

When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
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1answer
24 views

A continuous action of a compact group on a uniform space is equicontinuous?

I am stuck on how to prove the statement "A continuous action of a compact group $G$ on a uniform space $X$ is equicontinuous." So, essentially we want to show that for every entourage $\alpha$ of ...
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1answer
21 views

GNS-Construction: Involution

Given a C*-algebra $\mathcal{A}$. (It may or may not contain identity!) Consider a positive linear functional: $$\omega:\mathcal{A}\to\mathbb{C}:\quad A\geq0\implies \omega(A)\geq0$$ Construct its ...
4
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1answer
35 views

Spectrum of linear operator, essential spectral radius

Consider the operator $L:L^1(S^1)\to L^1(S^1)$ given by $$ (Tf)(x)=\dfrac{1}{2}\left( f\left( \dfrac{x}{2}\mod 1\right)+f\left( \dfrac{x+1}{2} \mod 1 \right) \right) $$ where we identified $S^1$ with ...
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1answer
46 views
+50

Representation of the Fréchet derivative of $〈f,e_n〉$, where $f:H→H$, $H$ is a Hilbert space and $(e_n)_{n∈ℕ}$ is an orthonormal basis of $H$

Let $H$ be a $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $f:H\to H$ be Fréchet differentiable and $$f_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$ ...
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24 views

space that maps time to continuous functions

I am learning the space that maps time into Banach spaces and I have the following confusion. Let $X=C_b([0,T]\times \mathbb{R})$ be the space of bounded continuous function $f(t,x)$, where the ...
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21 views

Find the norm of a linear functional in $L^2[0,1]$.

Define the linear functional $f : L^2[0,1] \text{(As completion of $C[0,1]$, all the continuous complex-valued function )} \mapsto \mathbb{C}$ by $$f(\psi)=3\int_{0}^{1}\psi(t)dt + i\int_{0}^{1} ...
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0answers
15 views

Transformation of inner product of wave functions under transformation of metric

Assume that we have a wave function $\psi(x)$ in the coordinate system $x$ in the Hilbert space $H_1$. The inner product of two states $\psi_1$ and $\psi_2$ are given as ...
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convergence of a subsequence in c0 [closed]

Let $f_n= e_1+e_2+…+e_n$ where $e_n $ is the $n$th standard unit vectors in $c_0$ .show that $(f_n)$ for all $n$ belongs to $N$ is a bounded sequence which does not have any weakly convergent ...
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1answer
16 views

Characterization of sequence in $l^1$

Consider the space of integrable sequence $\ell^1(\mathbb{R}) = \left\{ u \in \mathbb{R}^\mathbb{N} \ |\ \sum_{n=1}^\infty |u_n| < \infty \right\}$. I wonder if it possible that $$ \forall u \in ...
2
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1answer
64 views

Inequality on a general convex normed space

Assume $(X,\|\cdot\|)$ is a normed space with the following property: if $x \neq y \in X$ have norm 1 then $\|\frac{x+y}{2}\|<1$. (We then say that $X$ is strictly convex) Prove that if $C$ is a ...
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0answers
43 views

How is the MIT OCW set of PDE notes, compared with a graduate introductory course in PDE? [closed]

Here is the link: http://ocw.mit.edu/courses/mathematics/18-152-introduction-to-partial-differential-equations-fall-2011/lecture-notes/ The standard graduate introductory text is the one by Evans. ...
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22 views

Some question about polar set in functional analysis? [closed]

How to prove the propositions $b)$ and $d)$ displayed in the image about polar set?
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1answer
31 views

Question related to differentiable functions on Banach spaces

There is an interesting exercise on my Analysis book that I have not been able to solve: Let $\mathbb{E,F}$ be Banach spaces, $f:\mathbb{E}\to\mathbb{F}$ of type $\mathcal{C}^k$, $k\geq1$. Asume ...
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Given a closed subspace $Y$ of a Banach space $X$, $\mathrm{dim}(X/Y)' = \mathrm{dim}(X/Y)$

I am reading Peter D. Lax's functional analysis and have been stuck on his proof for the following theorem: Theorem. Let $C:X\rightarrow X$ be a compact map, and $T=I-C$. Then ...
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2answers
52 views

Alternative proof of Fundamental Lemma of Variational Calculus?

I am confused by one of the proof in the Calculus of Variations by Gelfand and Fomin. On page 9, we have Lemma: If $\alpha(x)$ is continuous on $[a,b]$, and if $\int_a^b \alpha(x)h(x)=0$ for every ...
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1answer
32 views

A convex subset of a Hilbert space

Assume $C$ is a convex subset of a Hilbert space $H$ ($C$ is not necessarily close) and $x_0\notin C$.Let $r=d(x_0,C)$. Prove that $\{y\in H\mid\|y-x_0\|\leq r\}\cap C$Has at most 1 element. I want ...
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2answers
170 views

Prove of inequality under a Hilbert space.

Let $x\neq y$ when $x,y\in H$ and H is a Hilbert space which satisfy $\|x\|=\|y\|=r$. Show that $\|\frac{x+y}{2}\|<r$. Actually in my question r=1 but as far as i could understand there is a way ...
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1answer
30 views

Norm equivalence on $l^1$.

Suppose that $\|\cdot\|$ is a norm on $l^1$ such that: a) $(l^1, \|\cdot\|)$ is a Banach space, b) for all $x \in l^1$ $\|x\|_{\infty} \leq \|x\|$. Prove that the norms $\|.\|$ and $\|.\|_1$ are ...
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0answers
19 views

Definition of higher order Fréchet derivative

My lecturer has recommended to us that we check that the obvious two candidates for the $k^{th}$ order Fréchet derivative are the same. That is defining the $k^{th}$ order Fréchet derivative ...
3
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1answer
36 views

is this an counterexample for: $(C[a,b],\| \cdot \|_2)$ is complete?

our prof wanted to show that $(C[0,1],\| \cdot \|_2)$ is not complete. So he said $$f_k(x) = x^k$$ is a counterexample. I wonder if this is true. I tried to show that $f_k$ is cauchy sequence. But i ...
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1answer
24 views

Resolvent $R(\lambda,A)x \to 0$ as $|\lambda| \to \infty$

If I have a closed operator $A:D(A) \to X$, not necessarily bounded on a Banach space $X$, and the resolvent is unbounded, can I show for a fixed $x \in X$ that $$R(\lambda,A)x \to 0$$ as ...
0
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1answer
25 views

Properties of Injective Operator on Hilbert Space

I am new to functional analysis and have the following issue: Given an infinite dimensional Hilbert space $H$ and an operator $f: H \times \Omega \to H$, where $\Omega$ is some finite dimensional ...
1
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1answer
37 views

The convex subbed of Hilbert space with no maximal norm

I wanted to show that if $\{e_n \mid n\in \mathbb{N}\}$ is an orthonormal basis for Hilbert space $H$ and put $$C= \left\{x\in H \mid \sum_{n \in \mathbb{N}} \left(1 +\frac{1}{n} \right)^2 | ...
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0answers
13 views

The map induced by multiplication of characteristic function in Sobolev space?

Let $U$ be the unit open ball in $\mathbb{R}^4$. Let $\chi_U$ be the characteristic map of $U$. It's clear that $\chi_U$ induces a map $$\varphi_U : C_c^\infty \longrightarrow L^2 \ \text{by} \ f ...
0
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1answer
30 views

The intersection of closed subspaces of Hilbert space

I am trying to prove that if there exists a finite dimensional closed subspace $M$ of a Hilbert space (say $X$) such that the intersection of the orthogonal complement of $M$ with another closed ...