Questions tagged [functional-analysis]
Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).
23 questions from the last 7 days
3
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136
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Murphy's Theorem 3.3.2 on positive linear functionals
The first part of Thm 3.3.2 in Murphy's book says
If $\tau$ is a positive linear functional on a $C^*$-algebra $A$, then $\tau(a^*)=\overline{\tau(a)}$ for all $a\in A$.
His proof makes use of an ...
- 2,982
3
votes
1
answer
104
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Radon measures in the book by Adams & Hedberg
I am currently trying to figure out what is a Radon measure in the book "Function spaces and potential theory" by Adams and Hedberg. Let me paraphrase the definition (Section 1.1.3 on page 2)...
- 33.9k
2
votes
0
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168
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Elements belonging to $c_0$ but not to $l^p$
Edit Let $1<p<\infty$. Is there a property $P$ (depending on $p$) for sequences such that for every $(\mu_n)_{𝑛\in\Bbb N}\in c_0$ there are sequences of scalars $(a_n)_{n\in\Bbb N}$ and $(b_n)_{...
- 139
0
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1
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124
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Using self-adjoint operator to define Euclidean norm
I am trying to understand definition of Euclidean norm on a finite dimensional space, $\mathbb{R}^{n}$, denoted as $\mathbb{E}$. The dual space is denoted by $\mathbb{E}^{*}$.
In the attached ...
- 143
0
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1
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60
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Is $C^\infty(U)\cap L^\infty(U)\cap W^{k,p}(U)$ dense in $W^{k,p}(U)\cap L^\infty(U)$?
I'm interested in whether smooth bounded functions are dense in Sobolev spaces. Specifically, letting $U\subset \Bbb R^n$ be open and bounded, is $C^\infty(U)\cap L^\infty(U)\cap W^{k,p}(U)$ dense in $...
- 14.4k
2
votes
1
answer
73
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if a sequence is in $l^p$
if we have $\sum_{n=1}^\infty |\lambda_n|^q < \infty,$ we may inductively find
$1 = m_1 < m_2 < ...$ such that, defining
$\sigma_k = \{n \in \mathbb{N} : m_k \leq n < m_{k+1}\}$ for $k = 1,...
- 139
1
vote
1
answer
70
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Equivalence of two definitions of finite complex measures
I am trying to understand Remark 4.29 in Pavlov's paper. He defines measures as follows.
Definition 4.28. A (complex infinite) measure on an enhanced measurable space $(X,M,N)$ is a map $\mu:M' →\...
- 119
0
votes
0
answers
80
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Property Preserved by Invertible $\mathbb{R}$-Linear Homeomorphisms Between Topological Vector Spaces over the Reals.
Please see Folland's second edition of Real Analysis, Modern Techniques and Their Applications or Wikipedia's Topological Vector Spaces for the definition of a Topological Vector Space over $\mathbb{R}...
- 237
2
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1
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54
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Question on controlling norm of output of a semidefinite operator by inner product
I'm asked to prove the following proposition:
Prop. If $H$ is a complex Hilbert space, $A\in L(H)$ satisfies that
$$
(Ax|x)_H\ge0,\ \forall x\in H,
$$
then
$$
\|Ax\|_H^2\le\|A\|_{H\rightarrow H}(Ax|x)...
- 375
0
votes
0
answers
84
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Continuous functional calculus (Modify the abuse of notations)
The proof of the theorem below contains several abuses of notation, which makes me quite confused sometimes. I would like a clear way of stating them, so I highlighted parts that needs clarification.
...
- 1,301
0
votes
1
answer
49
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$M$-modules of finite $M$-dimension
On page 25 of Introduction to Subfactors by V. Jones and V.S. Sunder we find the following definition:
Let $\mathcal{H}$ denote an arbitrary separable module over a II$_{1}$ factor $M$ with separable ...
- 1,150
0
votes
1
answer
66
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Exercise 10 of chapter 1 of Rudin's Functional Analysis [closed]
$ X $ and $ Y $ are topological vector spaces, and $ f $ is a linear map from $ X $ onto $ Y $ with $ \dim Y $ finite. The conclusion is that $ f $ is open.
I don’t know because it doesn’t allow me ...
- 1
2
votes
1
answer
108
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Is there a dense set of Lipschitz functions on the unit ball which all peak at the same point on the boundary?
I posted this same question over 2 years ago on MO but didn't get any comments or answers. The question is admittedly quite narrow, but I am still very interested in its resolution.
Let $U$ be the ...
- 1,247
0
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0
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49
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There are two definition of resolvent set
Recently, when I studied the spectrum of an operator, I came across two definitions of the resolvent set. In Kreyszig’s book, the definition is as follows:
Let $X \ne {0}$ be a complex normed space ...
- 219
0
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0
answers
49
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Central Limit Theorem via Fixed Point Theorem and Entropy
I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is:
The average $\bar{X} = \frac{1}{n} \...
- 1,987