# Tagged Questions

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### When is a vector space of polynomials dense in $C([0,1])$?

Weierstrass' theorem states, in particular, that the set of polynomials with real coefficients is dense (with the supremum norm) in the set of continuous function on $[0,1]$. Using the ...
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### Linear operator in $\ell^2$

Let $A \colon \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ be the linear operator defined by $\left( Ax \right)_k = \sum_{i \in \mathbb{Z}}a_{ki}x_i$, where $a_{ki} = 1/(k-i)^2$ if $k \neq i$ and ...
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### Small $\ell^p$ spaces are obtainable from $L^p$

I've seen that in a lot of books there is written that $$l^p=L^p(X,\Sigma,\mu),$$ where $X=\Bbb N, \Sigma=P(\Bbb N), \mu=\#$, ($\#$ is the counting measure). I would like to see how to prove it, ...
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### Books with exercises or problems on the space of functions of bounded variation

I am studying BV space (the space of functions of bounded variation) by using Evans & Gariepy's book. However, there are no exercises for BV space, and I have no idea where I can find some. ...
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### Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
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### If $f(x)g(y)$ is a measurable function, and $f$, $g \in L^{1}(dm)$, does this imply $g(y - x) \in L^{1}(dm)$?

Question rephrased Suppose we are working in $(\mathbb{R}, \Sigma(m^{*}) \times \Sigma(m^{*}), m \times m)$ where $m$ is Lebesgue measure. Note that our $\sigma$-algebra is not necessarily complete. ...
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### Question about statement of Fubini's theorem

This is a question on the statement of Fubini's theorem for measurable sets. The theorem looks like this: Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda = \mu \times \nu)$ be a complete ...
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### If $a_n\ge nb_n$ and the sequence $(b_n)$ is unbounded, then the differences $a_{n+1}-a_n$ are also unbounded

Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq1}$ be sequences of positive numbers such that $a_n\geq n b_n$ for all $n >1$. Prove that if $(a_n)_{n\geq 1}$ is increasing and $(b_n)_{n\geq 1}$ is ...
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### If $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, does that imply $(X, \Sigma, \mu)$ is $\sigma$-finite?

I'm having trouble proving or disproving the statement: If the product space $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, then so is $(X, \Sigma, \mu)$. I ...
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### What exactly is a product measure?

If we have $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ a complete measure space with underlying complete spaces $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$, and $\lambda = \mu \times \nu$, what ...
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### If $f \in L^{1}(d\mu)$ is nonnegative, can we conclude $\mu( \{ x \mid f(x) \neq 0 \} ) < \infty$?

I am trying to prove a statement, and I need the fact that: If $f \in L^{1}(d\mu)$ is a nonnegative function, then this implies $\mu( \{x \mid f(x) \neq 0 \} ) < \infty$. But I don't know ...
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### Integrable function with given condition is in $L^p$

Suppose $f:\Bbb R \to \Bbb R$ is integrable and there exist constant $c\gt 0$ and $\alpha \in (0,1)$ such $$\int_A |f(x)|dx\le cm(A)^\alpha$$ for every Borel measurable set $A\subset \Bbb R,$ where ...
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### Showing a sequence is in $\ell^2$ [duplicate]

I am working on the following problem. Suppose that $\{a_j\}_{j=1}^{\infty}$ is a sequence with the property that, whenever $\{b_j\}_{j=1}^{\infty} \in \ell^2$, one has ...
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### Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
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### Compact closure in $C([0,2])$

a) Does the closure of $\left\{f_n(x)=\sin(x^n):n=1,2,3\dots\right\}$ form the a compact subset of $C([0,2])?$ b) Does the closure of $\left\{f_n(x)=\sin(x^\frac1n):n=1,2,3\dots\right\}$ form the a ...
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### Show that C is a closed convex subset and its element of minimum norm

I have a lot of problems with the following exercise that I can't solve. Let $(L^1((0,1)), \|\cdot\|_{L^1}):=(E,\|\cdot\|)$ and $$C:=\{u\in E:u\geq0 \text{ a.e } x\in (0,1),\quad T(u)\geq 1\},$$ ...
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### Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$\check{\hat{f}}=\hat{\check{f}},$$ where $$\hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x$$ and ...
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### If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
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### For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
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### Fubini's theorem and $\sigma$-finiteness?

I'm reviewing my analysis notes, and I am really confused about what is meant by $\sigma$-finiteness being a hidden hypothesis of Fubini's theorem. Here is Fubini's theorem as was stated to me: ...
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I would like to know if this exercise is correct. Let $\Bbb R^\infty=\{x:\Bbb N\rightarrow \Bbb R: \exists n \text{ such that}\quad x(k)=0 \quad \forall k\geq n\}$. Show that $(\Bbb R^\infty, \| ... 1answer 56 views ### Selfadjoint Restrictions of Legendre Operator$-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$Problem: Let$Lf =-((1-x^{2})f')'$be the Legendre differential operator defined on the domain$\mathcal{D}(L)$consisting of twice absolutely continuous functions on$(-1,1)$for which$f, Lf \in ...
Let $a,b,c \in R$ Let D = {$(x_1,x_2,x_3): x_1^2 + x_2^2 +x_3^2 \leq 1$}. Let E = {$(x_1,x_2,x_3): \frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} + \frac{x_3^2}{c^2} \leq 1$} and ...