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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Open mapping lemma - are these versions equivalent?

Here is a version the Open Mapping Lemma given in class : Let $X$ be a Banach space and $Y$ be a normed space. Let $T : X\rightarrow Y$ be a bounded linear map. Assume there exist $M \geq 0$ and ...
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1answer
19 views

Topological Spaces: Uniform Structures

Disclaimer This thread is meant to record. See: Answer own Question Reference It is a follow-up to: Uniform Spaces: Neighborhood System It has relevance to: TVS: Uniform Structure Problem Given ...
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1answer
35 views

Spectrum of left shift operator: take two

This is my second attempt at calculating the spectrum of the left shift operator on a Hilbert space. I got stuck again and I would be grateful if someone could help. (You can find my previous (failed) ...
3
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0answers
13 views

Dual of $l^p$ Direct sum

I am asked to show that the $l^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $l^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ ...
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0answers
23 views

Prove that $T^*$ is injective iff $ImT$ Is dense

Let X,Y be two normed spaces, and $T:X\rightarrow Y$ a bounded linear operator. prove that the adjoint operator $T^*$ ($T^*f(x)=f(Tx)$ is injective iff $ImT$ is dense any help would be great guys. I ...
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1answer
36 views

Proving existence of a linear functional

Let $(X, \| \cdot \|)$ be a normed space, and let $A, B ⊂ X$ be disjoint convex sets such that $B$ is closed and $A$ is compact. Prove that there exists $\varphi ∈ X^*$ such that $$\sup_{a\in A} ...
3
votes
1answer
33 views

Show that an operator is well-defined

Let $v\in B(H)$, Define $u:|v|H\to H$ such that $u(|v|\xi) = v\xi$ . To show the map $u$ is well-defined, the author writes $$\||v|\xi\|^2=\langle v^*v\xi,\xi\rangle = \|v\xi\|^2$$ But I do not know ...
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0answers
18 views

Error in the calulation of the spectrum of the image of right shift operator in the Calkin algebra

If $S \in \mathcal{B}(\ell^2(\mathbb{N}))$ is the right shift operator $$ S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots),$$ and $\mathcal{C} := \mathcal{B}(\ell^2(\mathbb{N}))/\mathcal{K}$ is the Calkin ...
2
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1answer
18 views

Partial isometry and projection

The following is a Theorem of Murphy's C*-algebras and operator theory: Let $H_1, H_2$ be Hilbert spaces and $u\in B(H_1,H_2)$. If $u^*u$ is a projection, then $uu^*u=u$. To show it, for $\xi\in ...
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1answer
17 views

Showing that $a(\cdot,\cdot)$ is coercive

I am working on a problem and I have the weak formulation of Poisson's problem in $2$ spatial dimensions i.e. $u = u(x,y)$: $$a(u,v) = \ell(v) $$ where $$a(u,v)=\int_{\Omega}\nabla u\nabla ...
3
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1answer
14 views

A question about essential ideal

Let $I$ be a nonunital C*-algebra and $I\subset B(H)$ be any nondegenerate representation and define $$M(I)=\{T\in B(H): Tx\in I~and ~xT\in I, ~for ~all~ x\in I\}.$$ Then, how to prove $I$ ...
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1answer
38 views

Inverse in a functional space

I would like to understand why the inverse of a bounded operator must to be bounded too? In other context, all bijective function have an inverse but when we deal with a bounded operator it have to be ...
2
votes
2answers
44 views

Equality of two operators

The following is a fact in Murphy's C*-algebras and operator theory page 49: Suppose $u,v \in B(H)$, where $H$ is a Hilbert space, then $u=v$ if and only if $\langle u\xi,\xi\rangle = \langle ...
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1answer
55 views

Inequality on matrix norm: $ \lVert A^n \rVert \leq \lVert A \rVert^n $ [on hold]

If $A$ is a $n \times n$ matrix and assume we have a matrix norm $\lVert \cdot \rVert$. In a proof I need the following property: $ \lVert A^n \rVert \leq \lVert A \rVert^n $. I don't know how to ...
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0answers
20 views

Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$ I've been ...
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0answers
26 views

Is $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$ dense in $H^1(\Omega)$?

Can it be true that the space $$\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$$ is dense in $H^1(\Omega)$? If so, please give me a reference to this. Every $u \in H^1$ has $\Delta u \in ...
4
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1answer
44 views

When does a continuous function defined on a non-compact closed and bounded convex set has a fixed point?

Is there any result in fixed point theory which will give the existence of a fixed point for a continuous function defined on a non-compact, closed and bounded convex set?
3
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1answer
14 views

operators on Hilbert spaces have adjoints

The following is a Theorem of Murphy's C*-algebras and operator theory: In the last line of proof, he claims $u^*$ is linear, but I think it's conjugate linear because for $y_2,x_2\in H_2$, $x_1\in ...
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6answers
527 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
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0answers
21 views

a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ and Kolmogorov-Riesz compactness theorem

Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^{d}$ , $\mathcal{F}$ a set of all probability densities $f$ such that $\mathcal{F}$ is a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ ...
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1answer
43 views

What is the 'largest' space of integrable functions which is also a Hilbert space?

It is well known that $L^2(X,\mu)$, the set of functions $f:X \rightarrow \mathbb{C}$ such that $\int_X |f|^2 \text{d} \mu < \infty$, is a Hilbert space. Is there a Hilbert space $H$ such that ...
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0answers
19 views

Is the Fourier transform a tame linear operator?

$\mathcal{F}:C^{\infty}_{0}(B^d)\to L_{1}^{\infty}(\mathbb{R}^{d},\mu,w)$ $\mathcal{F}(f)=\hat{f}$ I'd like show that $\left\|\mathcal{F}(f)\right\|_{n}\leq\left\| f ...
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0answers
28 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
2
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2answers
35 views

extend a linear function

Let $P$ denote the subspace of $C^0([0,1])$ defined by polynomials restricted to [0,1]. Suppose that $l:P\rightarrow \mathbb{R}$ is a linear function with the property that $p(x)\geq 0$ in $x\in ...
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1answer
43 views

An unusual two dimensional sum

Can anyone prove or reference a proof for the following bound (unless it's not true!) $$\sum_{|\underline{k}|_{\infty} > M} \frac{1}{((k_1)^2 + (k_2)^2 )^2} \leq \frac{C}{M^2}$$ where ...
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0answers
31 views

Commutativity and norms of specific operators (Problem 2.7.10 in Kreyszig's functional analysis book)

This is Problem 2.7.10 from Erwin Kreyszig's Introductory Functional Analysis with Applications. Let $C[0,1]$ denote the normed space of all (real- or complex-valued) functions defined and ...
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0answers
35 views

Intersection of the closure of epigraph and subgraph of f function

Can you give an example of a function $f:R^p \to\ R$ such that $Grf \neq \overline{epif}\cap\overline{subf}$ where $p\in N$ and $Grf=\{(x,y)\in R^px R \mid f(x)=y \}, epif=\{(x,y)\in R^pxR \mid ...
2
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1answer
34 views

Convergence implies Abel summability, and we only need to consider when $s=0$?

Suppose $\displaystyle c_n\in\mathbb{C}\textrm{ and}\sum_{n=1}^{\infty}c_n=s$. Then, prove $\displaystyle\lim_{r\to 1^{-}}\sum_{n=1}^{\infty}r^{n}c_n=s$. In my text, the author hinted that: we only ...
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0answers
28 views

Boundedness of linear operator and weak convergence

Let us assume that $X,Y$ are Banach spaces and $T : X \to Y$ is a linear operator. Show that: $T$ is bounded $\Leftrightarrow$ for any sequence $ \{ x_n \}^{\infty}_{n=1} $ weakly convergent to some ...
2
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0answers
59 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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0answers
38 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
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0answers
39 views

Semigroup of operators: weak continuity at 0+ implies weak continuity at any t > 0

Let ($E$, $d$) be a metric space. Consider the semigroup $\{P(t)\}_{t\geq 0}$ of bounded linear operators on the Banach space $\hat{C}(E)$ of continuous real functions on ($E$, $d$) vanishing at ...
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0answers
10 views

weak* convergence of periodically extended function

Let $U := \Pi^d_{i=1}(a_i,b_i) \subset \mathbb{R}^d \ (a_i<b_i \ \text{for each} \ i )$ and let $f \in L^p(U)$ for some $1<p<+\infty$. Let us extend $f$ periodically on whole $\mathbb{R}^d$ ...
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0answers
24 views

weak* convergence for sequence in $ L^\infty$

Let $ \Omega \subset \mathbb{R}^d $ be a bounded and open set. Suppose $ \{f_n\} \subset L^{\infty} (\Omega) , f \in L^{\infty} (\Omega) $. Prove that $ f_n \rightharpoonup^* f \ \ \text{in} \ \ ...
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0answers
35 views

Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
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56 views

What is the “actual definition” of the following?

Imagine you were standing on the ground and as the ground starts moving you stay exactly where you are. Your change in movement, by standing on a moving surface, is like a path-line. Suppose we take ...
13
votes
3answers
276 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
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1answer
29 views

how to find norm of the operator Ax(t) = cos (t)*x(t)?

Please explain me how can I get norm of this operator: $Ax(t)=\cos(t)x(t)$ where $A\colon C[-\pi/2,\pi/2] \to C[-\pi/2,\pi/2]$. Thanks
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3answers
38 views

Borel measurable functions $f:[0,1]\to[0,\infty)$ which cannot be expressed as pointwise limit of nondecreasing sequence of step functions

An interval in this problem may be open, closed or half open. We regard a singleton $\{a\}$ as being an interval also. A step function is a real valued function on $\mathbb{R}$ which is a linear ...
1
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1answer
36 views

Trace map from $H^1$ into $H^{\frac 12}$, does this statement imply another?

Consider trace map $T:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ on a sufficiently smooth domain $\Omega$. It has a partial inverse $E$. If we have the statement $$F(u,Eu) = 0\quad\text{for all ...
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1answer
88 views

Determinant: Continuity

Reference Build-up on: Determinant: Definition Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its determinant $\det:\mathcal{L}(V)\to\mathbb{C}$. Introduce a norm ...
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32 views

C*-algebras: States?

I'd like to better understand states on C*-algebras. What properties should I investigate and in which order? (Positive functionals, extremal states, Schwarz's inequality, Kadison's inequality, what ...
3
votes
1answer
43 views

Help with proof that $H^{1}(\mathbb{R})$ is closed under multiplication.

Edit: Prove that if $u,v \in H^{1}(\mathbb{R})$ then $uv \in H^{1}(\mathbb{R})$. My idea is to approximate with functions in $C^{\infty}(\mathbb{R})$ with compact support. Let $u,v \in ...
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0answers
31 views

$c_o$ is not isometric to $c_0 \oplus c_0$

$c_0$ is the Banach space of sequences converging to zero and $c_0 \oplus c_0$ is its algebraical direct sum with itself equipped with the norm $||(\xi,\eta)|| := ||\xi||+||\eta||$. How to prove that ...
2
votes
1answer
31 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...
3
votes
4answers
182 views

Determinant: Alternative Definition (Matrices)

Reference Foundation for: Determinant: Continuity Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. The rank of an endomorphism: ...
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0answers
18 views

Density and Fredholmness

Let $X$ be a Banach Space and $Y$ a dense subset of $X$. An operator $T:X \to X$ is said to be Fredholm if it has closed range, $\dim \ker(T)<\infty$ and $\dim coker(T) < \infty$. Here is my ...
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0answers
22 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
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0answers
24 views

The dual of the space $L^\infty$. [duplicate]

As we know the dual of $L^p$s are $L^q$s where $\frac{1}{p} + \frac{1}{q} =1$, and dual of $L^1$ is $L^\infty$. What is dual of the space $L^\infty (E)$ where E is a measurable subset of $\mathbb{R} ^ ...
0
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2answers
26 views

Problem 8, Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...