# Tagged Questions

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### Borel measurable function

I'm struggling on the following question from a past paper: Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is a Borel measurable function and let $h:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined ...
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### Norm of multiplication operator

I have that $(X,\Omega,\mu)$ is a sigma finite space, and I have that $g$ is a measurable function. Assume that $fg\in L^p$ for all $1\leq p\leq \infty$. I want to show that $g\in L^\infty$. My idea ...
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### $l^\infty(I)$ and $l^\infty(J)$ isometrically isomorphic with $|I| \not= |J|.$

Is it possible for $l^\infty (I)$ and $l^{\infty} (J)$ to be isometrically isomorphic with the cardinality of $I$ not equal to the cardinality of $J$? I'm able to show that if $1\le p < \infty,$ ...
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### Sequence of finite dimensional subspace of a separable Hilbert space

This question is from an online course. I am not sure how to approach the problem. Let $H$ be a separable Hilbert space with a norm defined by the inner product. Let $(x_n)$ be a sequence in $H$ ...
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### Dual space of a normed space

The following is Exercise 3.5.4 of conway's Functional Analysis; Let $\{X_i\}_{i\in I}$ be a collection of normed spaces. If $1\leq p< \infty$ , show that the dual space of $\oplus_{p} X_i$ is ...
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### Regularity of Daubechies wavelet

I am reading the book Wavelets: Theory and applications by A. K. Louis, D. Maass, A. Rieder ...
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### Infinite Dimensional Vector Space: Finite Dim Subspace Closed and Nowhere Dense

Show that any finite-dimensional subspace $(S,\|\cdot\|)$ of an infinite-dimensional normed vector space $(V,\|\cdot\|)$ is closed and nowhere dense. Proof: Let $\{x^{(n)}\}_{n\geq1}$ be a ...
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### Simple question on integral

why: $$\int_0^1\int_{u(t)}^{u(t)+w(t)} f(t,v(t)) dv dt = \int_0^1 f(t,u(t)+\theta w(t)) dt; ~~\theta\in[0,1]$$ how to get this ? Please help me Thank you.
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### Annihilator in dual vector space

If $y(x) = x_1 +x_2 + x_3 = 0$ whenever $x = (x_1,x_2,x_3)$ is a vector in $\mathbb C^3$, then $y$ is a linear functional on $\mathbb C^3$; find a basis of the subspace consisting of all those vectors ...
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### Continuous linear functional

I want to show that $f:(\ell^1,\parallel. \parallel_1)\to \mathbb K$ defined by $f((x_n))=\sum\limits_{n=1}^{\infty}\dfrac{\vert x_n\vert}{n}$ is continuous linear functional and the norm of $f$ is ...
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### Bounded Inverse Theorem

$A$ is a bounded linear operator from $X$ to $Y$ (both Banach spaces). Show that if there exists $k > 0$ such that $\|Ax\| \geq k\|x\|$, for all $x$ then $\operatorname{range}(A)\,$ is closed. My ...
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### Weak Lower Semicontinuity Generalized to any $L_{p}$ space

I am having difficulty with the following proof: Generalize the weak lower semi-continuity of$L^{p}$ norms to all $1\leq p < \infty$; i.e., show that if $u_{n}\to u$ weakly in $L^{p}$, then ...
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### Range of a continuous linear mapping

I want to show that the range of the linear map $T:(\ell^1,\parallel .\parallel_1)\to (\ell^2,\parallel .\parallel_2)$ defined by $Tx=x$ is not closed. I considered a sequence $(x^{(n)})$ in ...
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### calculation of Simple terms

If $a>0$, $b\ne 0$ and $c \ge 0$ for which $\lambda$, $e^{\lambda x}[a\lambda^2-b\lambda-c]+ce^\lambda\ge1$ I really need this for some proof. $x\in[0,1]$ If I obtain $\lambda$ I can complete the ...
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### two-point boundary value problem for elliptic equations (ODE)

we consider two-point boundary value problem $$Au=-au''+bu'+cu=f~~~~~~~~~~~~~~~~ in ~~\Omega=(0,1)$$ $$u(0)=u_0,u(1)=u_1$$ where $a=a(x)>0$, $b=b(x)\ne 0$ and $c=c(x) \ge 0$ We must prove ...
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### Problem from Evans' PDE book, chapter 5, problem 5

I'm taking my first theoretical math course in a year and am bashing my head against a rock with this problem. "The sets $U,V$ are open, with $V \subset \subset U$ (compactly contained). Show that ...
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### Derivative and Morse lemma

can someone explain me this writing ? this is from K.C Chang book's "Infinite Dimensional Morse Theory and Multiple Solution Problems" i don't understand how to find exactly (5.9) , what it means ...
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### Implicit function theorem and derivative (proof of splitting lemma)

I have this theorem with a part of the proof: And I have this question: why $\nabla\hat{\varphi}(w)=(I-Q) \varphi(w+g(w))$ ? Please help me. Thank you.
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### Rudin 13.3 zero operator as adjoint

For an assignment I have to show that exists a densely defined operator on a infinite dimensional separable Hilbert space, such that its adjoint is the zero operator on the zero subspace. To show ...
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### Show that $T_\lambda : C^0([0,1], \mathbb{R}) \to C^0[(0,1), \mathbb{R})$ is contractive.

Let the aplication $T_\lambda : C^0([0,1], \mathbb{R}) \to C^0[(0,1), \mathbb{R})$ defined by $$T_\lambda \phi(x) = \lambda \int_0^1 \frac{x^2 + y^2}{1 + |\phi(y)|} \, dy$$ Show that $T_\lambda$ ...
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### In a Banach space X, its two Schauder bases have the same cardinal number?

The definition of Schauder basis is, there exist a set family F(whose cardinal number can be finite countable or uncountable), s.t. any x in X could be uniquely expressed countalbe linear combinations ...
### closed subspace $Y$ implies existence of non-zero linear functional $g$ such that $Y \subset \ker(g)$
I am working on an exercise and I am not sure if I am on the right track, so if anyone could give some hints I would be grateful. The exercise is If $Y$ is a proper closed subspace of $X$, prove that ...