Tagged Questions

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I have this, and I don't understand how to do the change of variable. Please help me Thank you.
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Prove that for every subspace we can find a finite number of linear functionals such that $W=\ker l_{1}\cap\cdots\cap \ker l_{k}$

In need of some assistance regarding this questions from a University textbook (I'm learning by myself). Its about Dual Spaces: Let there be $V$ a finite vector space (Has a basis) over $\mathbb{F}$. ...
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Help with Dual Spaces - Prove that either $w\in Im(f)$ **or** there exists ${l\in W^{*}}$ such that $f^{*}\left(l\right)=0$ **and** $l(w)=1$"

I'm in need of some assistance regarding this question. I'm learning Linear Algebra by myself using a university textbook and it has this question regarding Dual Spaces: "Let there be a linear map ...
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Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
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Is this proof valid for showing that this normed vector space is a Banach space?

A student suggested a way of proving that the following space defines a Banach space, but I'm not sure it can be done in this way. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Consider so the usual ...
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Question on the Deformation lemma

I have a functional $J$ defined on a hilbert space, with a finit number of a critical point $v_1,...,v_m$ let $b>\max\lbrace J(v_1),...,J(v_2)\rbrace$, and i want to prove that the set ...
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a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
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Orthogonal complement examples

I am looking for an example such that in a pre-Hilbert space $H$ we have for a subspace $U$ that (i) $\bar{U} \oplus U^\perp \neq H$ (ii) $\bar{U} \neq U^{\perp \perp}$ Since finite and closed ...
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Any positive linear functional $\phi$ on $\ell^\infty$ is a bounded linear operator and has $\|\phi \| = \phi((1,1,…))$

This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress. I'd like to prove that any linear functional $\phi$ on ...
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Showing that a map defined using the dual is a bounded linear operator from X' into X'

I have trouble answering the second part of the following exercise. Any help would be appreciated! Let $(X, \| \cdot \|)$ be a reflexive Banach space. Let $\{ T_n \}_{n = 1}^\infty$ be a sequence of ...
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Limits of a map on the space of bounded sequences

I'm working on an exercise consisting of several questions that I can't quite figure out. If someone could give me any tips I'd be very happy! For $\xi = (x_1, x_2,...) \in \ell^\infty$ define ...
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A m-dissipative operator is the generator of a $C_0$-semigroup

Let $(H;(\cdot,\cdot))$ be a complex Hilbert space and let $A: D(A) \rightarrow H$ be a densely dened closed operator on H. Let A be an m-dissipative operator that is \begin{align} &(Ax, x) \leq ...
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Is the canonical injection between $C^{k+1}[0, 1]$ and $C^k [0, 1]$ compact?

If $k=0$ by the Ascoli - Arzelá theorem the answer is yes, but i don't know how to proceed in the general case ($k> 0, k \in \mathbb{N}$). I tried to build a counter example using Riesz lemma ...
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A subbase for the weak topology of real functions

I'm reading a book chapter on weak topology, where it says that For a family $\mathscr{F}$ of real functions on $X$, the weak topology generated by $\mathscr{F}$ is denoted ...
convergence of product of sequences in $L^2$
Let $\displaystyle (f_n)_n \subset L^\infty ([0,1])$, $\displaystyle (g_n)_n \subset L^2 ([0,1])$ and $\displaystyle f \in L^\infty ([0,1])$, $\displaystyle g \in L^2 ([0,1])$ such that ...