Tagged Questions

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Definitions of hemicontinuity

can anyone see the equivalence or relation between the following two definitions of hemicontinuity that I encountered: Assume that $K$ is a closed, convex subset of Banach space $X$. Let $X^{*}$ be ...
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Solutions depending on something continuously

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find ...
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definition of unitary operator

Wiki says " A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $U^{*}U=UU^{*}=I$ , where $U^{*}$ is the adjoint of $U$, $I$ is the identity ...
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What's the name of this type operator?

If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$. If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ ...
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Characters only on commutative unital algebras?

I saw the following definition of a character: Let $A$ be a commutative unital Banach algebra. Then a non-zero homomorphism $\chi : A \to \mathbb C$ is called a character. For this definition to ...
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Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
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Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology ...
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The difference between hermitian, symmetric and self adjoint operators.

I am struggling with the concept of hermitian operators, symmetric operators and self adjoint operators. All of the relevant material seems quite self contradictory, and the only notes I have to go ...
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Orthogonal Projections: Definition + Characterization

Hey there I'm hanging at so many little steps when proving: $P\text{ orthogonal projection}\iff X\text{ orthogonal decomposable}$ My problem is there is simply missing a precise definition of what an ...
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Equivalent definitions for strictly positive elements

We have two usual definitions for strictly positive elements in C*-algebras: Let $A$ be a C*-algebra Definition (a) [MURPHY, C$^*$-algebras and Operator Theory] An element $a\in A_+$ is said to be ...
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A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
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Links between Minkowski metric, Hamming distance and Levenshtein distance

I know that Hamming distance is a particular case of Minkowski metric (with the specific definition of the subtraction). Also it seems that Hamming distance is a particular case of a Levenshtein ...
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Positive definite kernel vs. positive definite function

What is the difference between positive definite kernels and positive definite functions? As I understand it, a positive definite kernel is a positive definite function if it is translation ...
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How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
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What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
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Proper functions

I'm a bit confused with the definition of proper function, i.e.: "$f: X \to Y$ is proper if the inverse image of every compact set in $Y$ is compact in $X$." Can anyone provide a more intuitive ...
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Alternative definition of strong/weak operator topology.

Given two normed spaces $(X, ||\cdot||_X)$ and $(Y,||\cdot||_Y)$ the space of bounded linear maps $\mathcal{B}(X,Y)$ can be equipped with the strong operator topology (SOT) as follows: The ...
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Understanding Elliptic Operators

I don't have a strong background in partial differential equations so some of these questions might be quite basic. I first want to give some definitions which I am using. Definition: We define a ...
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Compact Operators and Complete Metrics Spaces

I have a couple of questions about compact operators and compactness in complete metric spaces: 1.I have the following implications: Let $Y$ be a metric space with $A$ a subset of $Y$. $A$ is ...
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Matrix with Functions as Entries

What do we call a matrix with functions as entries? $$\textbf{f(x)}=\begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix}$$
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Holder Space series term

Holder space $C^{k,\gamma}(\bar{U})$ has the norm $||u||_{C^{k,\gamma}(\bar{U})} := \sum_{|\gamma | \leq k}||D^{\alpha}u||_{C(\bar{U})} + \sum_{|\gamma|=k}[D^{\alpha}u]_{C^{0,\gamma}(\bar{U})}$, where ...
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What is a good definition of Hilbert space?

Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the ...
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I started studying functional analysis a couple of days ago, I have reached the Stone-Weierstrass theorem which is stated in my lecture notes as Let $X$ be a compact metric space, $A\subseteq ... 1answer 164 views What does “topological dual of a Banach space” mean? I am not sure what does the "topological" imply. Thanks. 1answer 83 views What is an operator norm? I came across this symbol where they are mentioning operator norms. I am not sure how it is different from the Frobenius norm. It was something like this:$|||\Omega-\hat{\Omega} |||_2$where ... 0answers 64 views definition of the Fourier transform of function on the sphere Let$f: S^{n-1}\longrightarrow R^n$be even continuous function. What is the Fourier transform of$f$? 1answer 185 views Definition of Hölder space I am wondering what is the definition for Hölder space$C^\gamma$when$\gamma\in \mathbb{N}$. Let's take the underlying field$\mathbb{R}^d$. Is it $$C^\gamma = \{f:\ f\in C^{\gamma-1}\}\cap \{f: ... 1answer 55 views If for every x_n such that x_n \rightarrow x, there exists a x_{n_k} such that Tx_{n_k} = Tx, is T continuous? Let X and Y be Banach spaces and T be the (possibly nonlinear) map T:X \rightarrow Y. T is continuous if for every x_n \in X such that x_n \rightarrow x, then Tx_n \rightarrow Tx. Is ... 1answer 79 views Good source for Triebel-Lizorkin spaces? I'm trying to look into different types of function spaces. At the moment, at least for function spaces involving integration, I only have L^p and W^{k,p}. The next function spaces I thought I'd ... 1answer 406 views Topology of uniform convergence? Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X,R) of real-valued continuous functions on X, with the topology of uniform convergence I am having a hard ... 1answer 889 views Semi-Norms and the Definition of the Weak Topology When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ... 1answer 457 views Functional independence Definition confusion: I wish to show that$$f(x,y)={-y\over x}$$and$$g(x,y)=\log |x|$$are functionally independent on some domain. What does that mean? What do I have to show? And how does one ... 1answer 79 views analogue of diag operator for functions If x\in{\rm I\! R}^n, then diagonal matrix \mathop{\rm diag}(x) is a linear operator \mathop{\rm diag}(x): {\rm I\! R}^n \to {\rm I\! R}^n. I am curious if there is some analogy for infinite ... 1answer 102 views About the positivity of the inner product on L^2[0,1] My textbook on Hilbert space theory claims that the map$$\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}~\mathrm{d}x$$is an inner product on C[0,1]. But I am not sure whether ... 2answers 74 views Alternate definition for boundedness in a TVS Let X be a topological vector space over \mathbb R or \mathbb C. A subset B\subset X is defined to be bounded if for any open neighborhood N of 0 there is a number \lambda>0 ... 1answer 135 views What is meant by `element x\in H of minimal norm' I do not seek a proof of the following exercise. I just want to understand this question in order to solve it myself. Let H be a Hilbert space over \mathbb R and let a, b\in H be such that ... 2answers 635 views On the definition of the Hausdorff distance \newcommand{\dist}{\mathrm{dist}\,} Let M be a metric space and \emptyset\neq A,B\subset M bounded closed subsets. The Hausdorff distance is defined as$$h(A,B)=\max\{\dist(A,B),\dist(B,A)\},$$... 1answer 225 views What is a differentiable functional? I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ... 2answers 540 views Domain of an operator in functional analysis I used to think that if we say$f$is a function from a set$X$to$Y$then this implied that$f$was defined on all of$X$. Because the definition of function is that it's a set$\{(x,y) \mid \text{ ...
The following is an example in my lecture notes: "Let $X$ be a locally compact topological space (that is, a topological space in which every point has a compact neighborhood). Then \$C_0(X)=\{f \in ...