# Tagged Questions

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### Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
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### Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
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### Boundedness of a closed operator

Can I get any help with this problem: Let $X, Y$ be Banach spaces, let $D$ be a subspace of $X$, and let $A \colon D \to Y$ be a closed linear operator. If $D$ is a closed subspace of $X$, ...
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### Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
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Let $H$ be a hilbert space and $T$ a compact self-adjoint operator on it. T is also injective on a dense subspace $U \subset H$ and we also have that $T(H) \subset U$. Now I am asked whether it is ...
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### Compact kernel operator on $L^p$ space

Let $\displaystyle U_1 \subset \mathbb R^{n_1}$ and $\displaystyle U_2 \subset \mathbb R^{n_2}$ measurable sets, $\displaystyle 1 < p,q < \infty$ and consider the measurable function ...
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### Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
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### Linear Operator and isomorphism

I wanted to be sure about the following: Let's say we have vector spaces normed spaces $X$ and $Y$ and a linear operator $T:X \rightarrow Y$. My idea was to reduce the properties that I need to show ...
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### Question about step in proof of Schauder's theorem

The statement is the following: Let $X,Y$ be Banach spaces and $T:X \rightarrow Y$ be a continuous linear operator. Then is $T'$ compact iff $T$ is compact. I have already understood the implication ...
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### Troublesome proof in Functional Analysis with dual vector space

Greetings to all of you I have tried to prove the following theorem but I am having some troubles with it. Let $X$ be a separable normed space and $(x_n')$ a bounded sequence in $X'$, then there is a ...
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### Conjugate convex functions property

$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$ And I ...
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### Relationship between adjoint operators, trace-class operators, compact operators and density operators in Quantum-Mechanics

I don't know much about Functional Analysis, but I was wondering about the following: In Banach spaces it is possible to define for every continuous opertor $T:X \rightarrow Y$ an adjoint Operator ...
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### Positive unbounded operator with zero not as an eigenalue

I am currently doing Quantum Mechanics and I am supposed to show that zero is an eigenvalue of a positive operator. I have no knowledge of Functional Analysis at that kind of level, so I was wondering ...
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### Compact operator and limit

I was wondering about something related to compact operators. If we have a compact operator $T:X \mapsto Y$ and a bounded sequence $(x_n)n$, then we know that there is a convergent subsequence ...
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### Compactness of integral operator

I need some help with this exercise. Let $f\in C^0_b(R^2)$ and consider the operator $[T(v)](x)=\int_0^x f(x,y)v(y)dy$ for every $x\in R$. Is this a compact operator $T:C^0[0,1]\rightarrow C^1[0,1]$? ...
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### Are these linear operators continuous?

For every polynomial $p(t)= \sum_{k=0}^{n} a_k t^k$ we declare its norm by $||a_k||=\sum_{k=0}^{n}|a_k|$. Now, I am supposed to check whether these maps are continuous and in case that they are I ...
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### Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21 , in a note for professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the ...
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### Calculate the norm of this operator

$C[0,1]=\{ f : [0,1]\to [0,1], f$ continuous$\}$ $||f||_\infty=\max_{t\in [0,1]} |f(t)|$ $T:C[0,1]\to C[0,1]$ defined by $$(Tf)(t)=\int_0^1e^{s+t}f(s)ds$$ Find $||T||$ The usual way to do this ...
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### Norm space, linear operator exercise, help please!

$f \in L_2[a,b]$ $Uf(s):=\int_a^bk(s,t)f(t)dt$ $k(s,t):[a,b]^2\to R$ continuous. show 1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$ 2) $U$ is linear and ...
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Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of all real valued functions on $\mathbb{R}^n$ whose first $k$ weak derivatives are in $L^p(\mathbb{R}^n)$. Assume that $$\frac{1}{q} = \frac{1}{p} ... 1answer 50 views ### Restriction to \mathbb{R}^{d-1} as an operator on L^2(\mathbb{R}^d) Identify \mathbb{R}^{d-1} with \mathbb{R}^{d-1}\times \{0\}\subseteq \mathbb{R}^d. Is there a bounded operator T: L^2(\mathbb{R}^d)\rightarrow L^2(\mathbb{R}^{d-1}) such that T(\phi)=\phi ... 2answers 144 views ### Strong convergence of operators I'm working through the functional analysis book by Milman, Eidelman, and Tsolomitis, and I have a question. The book states a lemma that I'm a bit confused about: A sequence of operators T_n\in ... 0answers 49 views ### Relationship of two generalizations of the real/complex calculus On the one hand, one has the various functional calculi from Operator Algebras. The continuous functional calculus for C* algebras, the bounded borel functional calculus for Von Neumann Algebras, the ... 1answer 90 views ### Product and Quotient rule for Fréchet derivatives Does anyone know whether the product/quotient rule for FrÃ©chet derivatives still hold? For example, consider the evaluation operator:$$\rho_x : (C[a,b],\|\cdot\|_\infty) \rightarrow ...
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I am looking in the space of test functions $\{f \in C^\infty|f^{(n)}(a)=f^{(n)}(b)=0\};n \in \mathbb{N}_0\}$whether the n-th derivative is a self adjoint operator. the dot product is given by ...
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### Extension of a Bounded Operator on $L^p$ to $L^r$

Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a ...
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### Find norm of the integral operator

Find norm of the following bounded linear operator $$Ax(t)=\int_0^1e^{-ts}x(s)ds$$ where $x\in C[0,1]$ and $t\in[0,1]$. Please help me.
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### Prove that the sequence is in $\ell^{2}$. [duplicate]

Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$ What I've tried so far is ...
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### Hilbert's Inequality

Could you help me to show the following: The operator $$T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy$$ satisfies $$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p$$ for $1 <p< \infty$ where ...
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### Prove that the integral operator is bounded [duplicate]

Prove that the following operator is bounded on $L^{2}(0, \infty)$: $Af(x)$ = $\frac{1}{\pi} \int_{0}^{\infty} \frac{f(y)}{x+y}dy$ with $||A|| \le 1$. Attempt at Solution It can be shown that: ...
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### What does this phrase about the weak topology of bounded operators mean?

Can somenone remind me of the meaning of the following statement: the family of operator valued functions $A(\omega)$ converges to $A(\omega ')$ in the weak topology of bounded operators from ...
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### Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$\mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy$$ so that it has all its ...
There are some theorems that say in a unital C* algebra $A$ when one can deduce that the functional calculus of a continuous function f is continuous as map from some subset of $A$ to $A$. In the ...