All Questions
Tagged with lp-spaces hilbert-spaces
224
questions
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Spectrum of the laplacian outside of a compact
Let us consider $A$ a translation invariant lower semi-bounded operator on $L^2(\mathbb{R}^n)$ with domain $D(A)$ and with empty discrete spectrum (I exclude bound states). I have the following ...
- 57
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1
answer
31
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A Multiplication operator in a Hilbert space: $M_h$ is bounded and $||M_h|| \leq || h||_{\infty}$ [duplicate]
I'm trying to understand the example below, taken from Axler's Measure Integration and Real Analysis book.
How does one prove that $M_h$ is bounded and that $||M_h|| \leq || h||_{\infty}$?
I was ...
- 5,331
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Prove operator $(Tf)(x)=sin(x)\cdot f(x)$ is not compact
Given the following operator in $L_2[0,1]$
$$(Tf)(x):=sin(x)\cdot f(x)$$
Prove or Disprove that the opertor is Compact.
I thought it is compact and used arzelà–Ascoli theorem, but apparently I am ...
- 617
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1
answer
115
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Square integrable for universal approximation
Let's consider square-integrable functions $f \in L^2\left(I_n\right)$ with the definition of the $\textit{discriminatory}$:
$\textbf{Definition:}$. The activation function $\sigma$ is called ...
- 1,023
2
votes
3
answers
319
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How to compute the numerical radius of the right shift operator?
Let $T$ be the right shift operator on $\ell^2$ defined by $T(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)$.
The numerical radius of $T$ is defined by $w(T)=\sup\{|\langle Th,h\rangle|:\, \|h\|=1\}$. It is well ...
- 1,130
2
votes
1
answer
169
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Completion of some $C[0,1]$ functions with some inner product is a reproducing kernel Hilbert space
The problem
Let $X$ be the space of $C^1[0,1]$ functions with the special property $f(0) = 0$. Consider the inner product defined in the following way:
$$\langle f, g \rangle = \int_0^1 f'(x) \...
- 540
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57
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Is the set of polynomials of degree $\le n$ closed in $L^2(a,b)$? [duplicate]
Let $a<b$ be real numbers. I'm asked to prove that for every $f\in L^2(a,b)$ there exists a unique polynomial $p_n$ of degree less than or equal to $n$ such that
$$\|f-p\|_2\ge\|f-p_n\|_2,$$
for ...
- 872
1
vote
1
answer
155
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Show that $g_n(x)g_m(y)$ forms an orthonormal basis for $L^2(\Omega \times \Omega)$ when $g_n$ is an orthonormal basis of $L^2(\Omega)$
Let $\Omega \subset \mathbb{R}^n$. Show that $g_n(x)g_m(y)$ forms an orthonormal basis for $L^2(\Omega \times \Omega)$ for all $n,m \geq 1$ when $g_n:\Omega \rightarrow \mathbb{C}$ is an orthonormal ...
- 169
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58
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Showing that the integral operator $Ku(x):=\int_{A}k(x,y)u(y)dy$ is a Hilbert-Schmidt operator if $k\in L^2(A^2)$ for $A\subset\mathbb{R}$
Let $A\subset\mathbb{R}$ and $k\in L^2(A^2;\mathbb{C})$. I am trying to understand how we can conclude that the integral operator $K$ defined to act on $L^2(A;\mathbb{C})$ by $Ku(x) := \int_Ak(x,y)u(y)...
- 1,221
5
votes
1
answer
119
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Finding an explicit formula for the projection on a subspace of an Hilbert space
Given the Hilbert space $L^2([-1,1])$ endowed with the usual inner product, consider the following operator:
$$Tf:=\int_{-1}^1f(x)e^x \mathrm dx $$
Let $N:=\ker T$, find an explicit formula to find ...
- 5,840
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3
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115
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Brezis' exercise 8.28.4: how to prove that $\langle Tf, f \rangle \ge 0$?
Let $I$ be the open interval $(0, 1)$ and $H := L^2 (I)$ equipped with the usual inner product $\langle \cdot, \cdot \rangle$. Consider the linear map $T: H \to H$ defined by
$$
(Tf) (x) = \int_0^x t ...
- 17.6k
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1
answer
65
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Is the span of these vectors dense in $L^2(-1,1)$?
Let $T$ be the operator $Tf = xf$ on $L^2(-1,1)$. I am trying to figure out if any of $x, \mathrm{sgn}\ x$, or $x\mathrm{sgn}\ x$ is a cyclic vector.
I believe that $x$ is not, since the span of $\{x, ...
user1147800
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44
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Solution verification of continuity of a function in $L^p(\mathbb R)$
We are asked to prove the following:
Let $f\in L^p(\mathbb R)$, with $1\leq p<\infty$. Show that
$$g(x)=\int_{x}^{x+1}f(t)\,dt$$
is continuous. We did the following:
From Hölder's inequality, for $...
0
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37
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Orthonormal Basis of $L^2[0,1]$ having indicator functions?
I was asked to construct an ONB of $L^2[0,1]$ having functions taking at most two values. By suitably scaling, I can think them as indicator functions. So question boils down to finding countably many ...
2
votes
2
answers
88
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Prove that the $L_2$ norm of a function is equal to the sum of the squared Fourier series coefficients?
I am new to this sort of analysis, so corrections to my terminology or understanding are very welcome. In one of my courses, an equivalence is drawn between $l^2$ and $L^2$ spaces. For $(c_1, c_2, ...)...
