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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34 views

is this equality true?

Let $\{M_\alpha\}$ be a family of closed subspaces of a Hilbert space H. Is this equality true? ${(\cap M_\alpha^\bot)^\bot}=\overline{span(\cup M_\alpha)}$. thanks for your help.
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51 views

isometric embedding of l^2

CLAIM: Let $H$ be an infinite dimensional $\mathbb{R}$-Hilbert space. Then the $\ell^2$ sequence space can be embedded in $H$. I think it could be true since every Hilbert space has an orthonormal ...
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0answers
46 views

Does integration over continuous compactly supported functions completely determine the measure?

If $X$ is a locally compact space. And $\mu, \nu$ are two Radon measures on $X$. If for any continuous compactly supported function $f \in C_c ( X \to \mathbb R )$ we have $ \int f d\mu = \int f d\nu ...
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1answer
43 views

Question about double integral [closed]

how to do to transform this $$\displaystyle\int_0^t (\frac{1}{p(s)})(\int_0^s f(\xi,u(\xi))d\xi)ds $$ int one integral? and please why $\int_0^t\int_0^s y(\xi)d\xi =\int_0^t (t-s) y(s) ds$???? ...
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0answers
37 views

Separability of a certain space of continuous functions

Let $I$ be a separable, locally compact Hausdorff space, and let $V$ be a separable, locally convex, complete topological vector space. Consider the function space $C(I, V)$ with the compact-open ...
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28 views

Question about meaning of evolution problem.

Consider the following "evolution problem" $f(t) - u_t(t) \in \partial \psi(u(t))$ $u(0) = u_0$ Where $f:[0,T] \rightarrow H$ $ u:[0,T] \rightarrow H$ $ \psi:H \rightarrow (-\infty,\infty]$ is ...
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1answer
44 views

Show norms are equiv. on $C^1[a,b]$: $\Vert f\Vert _1=\Vert f \Vert_{\infty}+\Vert f' \Vert_{\infty},\Vert f \Vert_2=|f(a)|+\Vert f' \Vert_{\infty}$

Here is what I got as a proof. My question is at the end. Thanks On $ C^1[a,b]$ we have the norms $$\Vert f\Vert _1 = \Vert f \Vert_{\infty} + \Vert f' \Vert_{\infty},\quad \Vert f \Vert_2 = |f(a)| + ...
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1answer
22 views

Please verify my work about an equicontinuous sequence

Please check this work below. It is self-explanatory. I am unsure because I use a sequence composed with another sequence with the same index ($f_n^{-1}(u_n)$). We have a sequence of functions ...
4
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2answers
77 views

Bounded data means bounded solution to parabolic PDE

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider $$u_t - \Delta u = f$$ $$u|_{\partial\Omega} = 0$$ $$u(0) = u_0$$ or more generally replace $\Delta$ with a suitable ...
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1answer
53 views

Are these $f_n$ equicontinuous?

Let $f_n$ be a sequence of real-valued functions defined on $\mathbb{R}$ satisfying $f_n \to f$ uniformly in the compact subsets of $\mathbb{R}$ $f_n^{-1}$ is bi-Lipschitz $1 \leq (f_n^{-1})'(x) ...
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1answer
37 views

Converse of existing question on L^p convergence

My question is about this: Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$ It was shown that the author's question was indeed true by the use of MVT. Is the ...
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1answer
39 views

Is every closed set $K\subseteq \mathbb{C}$ the essential range of a measurable function?

For a complex-valued function $h$ on a measure space $(S,\Sigma, \mu)$, the $\textit{essential range}$ of $h$ is the set of all $\lambda \in \mathbb{C}$ such that for all $\epsilon >0$ the ...
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1answer
54 views

How are generalized frames related to biorthogonal bases?

How are generalized frames related to biorthogonal bases? It seems like frames are a possible solution if neither orthonormal nor biorthogonal bases are available. I thought the generalized frames ...
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0answers
51 views

If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
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1answer
46 views

Gronwall inequality for $\frac{d}{dt}u(t) \leq C_1u(t) + C_2\sqrt{u(t)}$

I have the inequality $$\frac{d}{dt}u(t) \leq C_1u(t) + C_2\sqrt{u(t)}$$ for a positive function $u$. Is there a Gronwall inequality that I can use to write $$u(t) \leq C_3u(0)?$$ or something ...
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1answer
98 views

Range and kernel of linear operators

I have a compact linear operator $T$, and I would like to show $$\operatorname{range}(\lambda I-T)=(\ker(\overline{\lambda}I-T^*))^\perp.$$ I have shown the forward inclusion "$\subset$" directly by ...
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2answers
161 views

Compact operator whose range is not closed

I am asked to find a compact operator (on a Hilbert space) whose range is not closed, but I am having trouble coming up with one. My guess is that you need to have some sequence in the range that ...
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1answer
114 views

Hardy-Littlewood-Sobolev fractional integration inequality fails at endpoints

Here's a version of the theorem: If $1 < p, r < \infty$ and $ 0 < \alpha < n $ be such that $ \frac{1}{p} + \frac{ \alpha }{ n} = \frac{1}{r} + 1 $. Then for any $ f \in L^p ( \mathbb R ^n ...
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0answers
30 views

Prove operator convergence

I have to proove that given $X$ a normed space and $Y$ a Banach space, if the sequence of bounden linear operators from $X$ to $Y$ $\{A_n\} \rightarrow A$ and the sequence $\{x_n\} \rightarrow x$ then ...
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1answer
56 views

Relations between normed spaces

Is the application $$ Id:( C([0,1]), \|\cdot\|_{\infty})\to ( C([0,1]), \|\cdot\|_{1}) $$ open? where $Id(f)=f$, $\|f\|_{\infty}=\sup\|f(x)\|$ and $\|f\|_1=\int |f(x)|dx$
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1answer
102 views

Does existence of a non-continuous linear functional depend on Axiom of Choice?

Well, it is easy to construct a non-continuous linear functional on an arbitrary infinite-dimensional vector space (assuming Choice, and taking a basis etc.). I think it is intuitive to say that: ...
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1answer
58 views

Vanishing of a quadratic form along the orbits of a unitary group

Let $H$ be a (complex) Hilbert space and let $B\colon H\times H\to \mathbb{R}$ be a continuous sesquilinear form (i.e. a continuous function that is linear in one argument and conjugate-linear in the ...
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1answer
65 views

Compactness of set of projections

In a proof I am reading I saw the statement that the set of projections onto one fixed finite dimensional subspace of a Banach space $X$ is compact. Is this obvious, for some reason I just don't see ...
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3answers
51 views

Examples of spectrum of compact operators on the sequence space $l_2$

Suppose $T$ is a compact operator on the sequence space $l_2$, and let $\sigma(T)$ be its spectrum. Is it possible to find a $T \ne 0$ such that $\sigma(T) = \{0\}$? Also, is it possible to find $T$ ...
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1answer
101 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
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0answers
35 views

What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
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1answer
26 views

Why is oblique projection not a self adjoint operator?

Why is oblique projection not a self adjoint operator? Here is an explanation of oblique projection.
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24 views

Is the composite of a state and a $\ast$-homomorphism again a state?

If $\cal{A,B}$ are unital $C^\ast$-algebras and $\varphi:\cal{A\to B}$ is an unital $\ast$-homomorphism. Then it is clear that $\rho\circ\varphi$ is a state on $\cal{A}$ for any state $\rho$ on ...
2
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1answer
57 views

lower semi continuous on Banach space implies locally bounded?

Let $(X, \|\cdot\|)$ be a Banach space; and $f:X\to [0, \infty)$ is lower- semi continuous on $X.$ My Question is: Can we expect $f$ is bounded in some open subset of $X$ ? [If answer is ...
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0answers
12 views

Particular form of states [duplicate]

On can prove that for a $C^*$-algebra $A$ and an irreducible representation $(\psi,H)$ of $A$ that for every unit vector of $H$ the state $t_x(a)=\langle\psi(a)x,x\rangle$ is pure. But now let $H$ be ...
1
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1answer
30 views

question involving $\ell^p$ spaces

I would like to prove the following: I am not asking for a solution! I would simply like a bit of guidance, I can't seem to get started on this problem. Would the proof involve using weak ...
5
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1answer
190 views

Showing that $\|f\|_p\to\|f\|_{\infty}$

I know this question has been asked a lot in this site, I've been checking those questions myself, however according to the theory we use in class there are things that I can't use and/or I don't know ...
0
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0answers
35 views

A Banach space $X$ is relfexive iff $X^*$ is reflexive. [duplicate]

I am trying to prove the following: A Banach space $X$ is reflexive iff $X^*$ is reflexive. Thus far, I have proven the forward direction: Let $J_X:X\mapsto X^{**}$ be the mapping defined by ...
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2answers
21 views

Is this function well defined for any $g\in L^q(\Omega)$?

If $p,q\in(1,\infty)$ such that $\frac 1q+\frac1p=1$, given $g\in L^q(\Omega)$ we difine: $$\Phi(g):L^p(\Omega)\to\Bbb R \\ \Phi(g)(f):= \int_{\Omega}fg$$ I know this is a basic question, but how do ...
1
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1answer
104 views

Show that $\log(x)$ is a Bounded Mean Oscillation (BMO)

As an extension of our class notes, we were asked to show that the function $w =\log(x)$ is a Bounded Mean Oscillation (BMO). First off, I believe our professor made a mistake, and really wanted us ...
1
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1answer
26 views

Bases in Hilbert space

There's a theorem that states that having Hilbert space $H$, orthonormal basis $\{x_n\}$, and a set of linearly independent unit vectors $\{y_n\}$, such that $\sum\limits_{n=1}^{\infty}\|x_n - ...
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1answer
38 views

Resolvent of the differentiation operator on the torus

Suppose we have the space $L^2(\mathbb T)$, that is, the space of periodic functions that are in $L^2$. Let $L^0$ be the operator of differentiation, i.e $L^0 f = f'$ where the domain of $L^0$ is the ...
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1answer
42 views

Proof: In a topological vector space, every neighborhood of $0$ contains a balanced neighborhood of $0$

I was reading this proof in Rudin 2/e (Th 1.14), but couldn't work it out. Suppose $U$ is a neighbhorhood of $0$ in the topological vector space $X$, then Since scalar multiplication is ...
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1answer
38 views

Examples of functions with values in distributions

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in ...
2
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1answer
47 views

Convergence of function in $L^1$ space

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
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2answers
67 views

Sequence of unit vectors in a Hilbert space

Question: Let $H$ be a Hilbert space and $\{\xi_{i}\}\subset H$ be a sequence of unit vectors. Suppose that $||T_{j}(\xi_{i})-\xi_{i}||\rightarrow0$ as $i\rightarrow\infty$, for $j=1, 2, ...n$ (here ...
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0answers
28 views

Question about Green function

how to find the Green function of this problem : $$ \begin{cases} -(p(t)u'(t))'=\lambda f(t,u(t)) ~ \text{a.e.} ~t>0\\ u(0)=u(+\infty)=0 \end{cases} $$ Thank you
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0answers
49 views

bilinear form, anti symmetric part

$\mathcal{H}$ : real Hilbert space with inner product $(\,,\,)$ and norm $||\,||:=(\,,\,)^{1/2}$ Let $D$ be a linear subspace of $\,\mathcal{H}$ and $\mathcal{E}$ : $D\times D\to \mathbb{R}$ a ...
1
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1answer
71 views

Are the following norms equivalent?

We have the norms $||f||_1=||f||_\infty+||f'||_\infty$ and $||f||_2=|f(a)|+||f'||_\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I prove/disprove this.
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1answer
43 views

Question about injection on an unbounded space

I have this space $$C_0((0,+\infty))=\left\lbrace u,u\in C((0,+\infty)),\lim_{t\rightarrow +\infty} u(t)=0\right\rbrace$$ with the norm $$||u||_{\infty}=\sup_{t\geq0}|u(t)|$$ how to prove that ...
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1answer
128 views

These germs make me sick!

I need a "mini-crashcours" concerning the space of germs of continuous functions in order to solve an exercise which requires me to show that limits in this space aren't always unique. We have ...
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1answer
73 views

Example of weak derivative on multivariable function

In order to explain about the concept of weak derivatives, I plan to give examples on them. I already manage one example for the single-variable case, but I think it would be better if I can provide ...
4
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3answers
104 views

Showing that a sequence converges in norm

I am trying to prove the following: Let $X$ be a normed linear space satisfying the property: $\forall \left\{x_n\right\}, \left\{y_n\right\} \subseteq X $, we have $\|x_n\|=\|y_n\|=1, ...
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2answers
61 views

showing uniqueness of a Hahn Banach extension

I am trying to prove the following: If $H$ is a Hilbert space and $G\subseteq H$ is a closed linear subspace, then any bounded linear functional on $G$ has a unique Hahn-Banach extension on $H$. So ...
2
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1answer
52 views

An exercise about projections on Hilbert space

Let $H$ be a Hilbert space with an orthonormal basis $\{v_{n}\}_{n=1}^{\infty}$. The C$^{*}$-algebra $K$, the set of all compact operators on $H$, is a non-unital C$^{*}$-algebra. Let $p_{n}$ be the ...