Functional analysis is 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, as well as measure, ...

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204 views

My proof of closed range theorem: where is the mistake?

I tried to prove the closed range theorem please could someone check my proof: Theorem: Let $H,H'$ be Hilbert spaces and $u\in B(H,H')$ and $u^\ast$ its adjoint. Then $\mathrm{im}(u)$ is closed if ...
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40 views

Causal differential equation with local Lipschitz condition

Consider the equation $$ u'(t) = (Fu)(t) $$ where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type) nonlinear operator. It means that the value of $(Fu)(t_0)$ ...
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1answer
68 views

Projection in Hilbert space onto non-closed subspace?

I know that if $H$ is a Hilbert space and $C$ a closed subspace one can define the orthogonal projection onto $C$ as the map $x \oplus y \in H = C \oplus C^\bot \mapsto x$. I am wondering: Is it ...
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81 views

Positive linear functional on a C*-algebra is bounded

The following is a theorem of Murphy's C*-algebras and operator theory: My question: I think in the proof of theorem, Murphy uses the assumption $|\tau(a)|<M$ for positive elements $a\in ...
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1answer
164 views

Question on extending densely defined linear operators on Hilbert space

Let $H$ be a Hilbert space. Is there a theorem that states that for a densely defined continuous linear operator $T: D(T) \subset H \to H$ there exists a unique continuous linear extension to ...
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181 views

Resolvent: Norm

Given a Banach space. Consider a closed operator: $$T:\mathcal{D}(T)\to E:\quad T=\overline{T}$$ Due to the Neumann series it holds: $$R(\lambda):=(\lambda- ...
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1answer
173 views

Proof a corollary on Hahn-Banach theorem

A corollary says Let $F\subset E,$ is a closed linear subspace of $E$, where $E$ is a normed vector space. Then it exists a linear functional $f\in E^*,f\neq 0$ such that $$ \langle x,f \rangle = 0, ...
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126 views

Intuitive understanding of the operator norm?

I understand various vector norms, but I don't understand operator norms. Specifically, norms on linear operators. Can anyone explain them?
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62 views

Strictly convex iff norm is strictly sub additive

Show that the closed unit ball in a normed linear space is strictly convex i ff the norm is strictly sub additive. One part is easy strictly sub additive implies strictly convex, but I'm not ...
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21 views

If $u_n \to u$ in $L^2(0,T;L^2(\Omega))$ and $f_n \to f$ uniformly, does $f_n(u_n) \to f(u)$ in $L^2(0,T;L^2(\Omega))$?

Let $\Omega$ be an unbounded domain. Suppose we have $u_n \to u$ in $L^2(0,T;L^2(\Omega))$. Let $f_n\colon \mathbb{R} \to \mathbb{R}$ be a sequence such that $f_n \to f$ uniformly. We know that ...
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56 views

what is$ f^{(n+1)}(x)$ as a function [closed]

Define $f^{(n+1)}(x)$ in function form. Is it $f(f^{(n)}(x))$ or is it $f^{(n)}(x)*f(x)$. Or is it something else completey. Thank you so much. I'm actually studying functions and this was something ...
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33 views

If $f_n \to f$ pointwise and $|f_n(x)| \leq C$ for $n$ large enough, does Dominated Convergence Theorem still work?

Let $f_n \to f$ pointwise a.e., and for a.e. $x$, we know $|f_n(x)| \leq C$ as long as $n$ is large enough (so if $n \geq N$ where $N$ may depend on $x$). If we are integrating over a compact domain, ...
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85 views

Are $L^\infty$ bounded functions compact in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a compact subset of $L^2$? (Compact in the topology induced by the ...
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2answers
75 views

Bounded Linear Operator and the Adjoint

Let $S$ be a linear operator with dense domain $\mathcal{D}(S)$ in the Hilbert space $\mathcal{H}$. Assume that the domain $\mathcal{D}(S)$ belongs to a larger domain, namely $\mathcal{D}(S) \subset ...
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64 views

Show that the space $C^{0, \gamma}(U)$ is complete

How can we show that the space $C^{0, \gamma}(U)$ is complete?? I have tried the following: So that the space is complete, the following has to stand: $$\forall \epsilon >0, \exists n_0 ...
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1answer
83 views

Prove $\exists x$ st $\|x\|=1$ and $\operatorname{dist}(x,Y)=1$

If $Y$ be a finite-dimensional proper subspace of a normed linear space $X$, then show that there exists $x\in X$ st $\|x\|=1$ and $\operatorname{dist}(x,Y)=1$ For any $x \in X$ st $\|x\|=1$ we ...
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44 views

A quick question on Signed measures and the Legesgue Decomposition Theorem

Let $(X,\mathcal{A},\mu)$ be a (positive) $\sigma$-finite measure space. Then the Lebgesgue Decomposition Theorem states if $\nu$ is a sigma finite measure (positive, signed or complex), then there ...
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1answer
91 views

Convergence in norm and in mean

Here is my problem: If $\{f_k\}$ is a sequence in $L^2$ and $f_k\to f$ in mean, show that $\{||f_k||_2\}$ is a bounded sequence of real numbers. Before I start doing the problem, I would like to know ...
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1answer
54 views

Does $f \in B[a,b]$ with only countable many points of discontinuity imply $f ∈ \overline{S[a,b]}$?

If $f$ $∈$ $B[a,b]$(the set of bounded functions on[a,b]) has only countable many points of discontinuity, does it follow that $f$ $∈$ $\overline{S[a,b]}$ where $S[a,b]$ denotes the collection of all ...
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101 views

Hilbert space structure on $C^{*}$ algebras

What is an example of an infinite dimensional $C^{*}$ algebra with a Hilbert space structure (not merely pre-hilbert structure) such that the orthogonal complement of each closed left ideal ...
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1answer
67 views

Normable LF space?

Let $E$ be a locally convex Hausdorff space and consider its topological dual $E'_b$ endowed with the strong dual topology, i.e. uniform convergence on all bounded sets of $E$. In Trèves, "Topological ...
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1answer
58 views

Characterization of lower semicontinuous functions by neighbourhoods

I tried several times to show the following thing: Let $f:X\to \mathbb R$ be a lower semicontinuous function. Then, $$(\forall \epsilon >0)\ (\forall x\in X)\ (\exists V\in\vartheta(x))\ s.t. ...
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49 views

A converse to the Lax Milgram theorem

Assume that a banach space $X$ satisfies the Lax Milgram theorem. Must $X$ be isomorphic to a Hilbert space?
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1answer
51 views

compactness in finite-dimensional space

I know that if the set $\{(x,y):\|x\|=\|y\|=1,\|x−y\|\ge\epsilon\}$ is compact,in a finite-dimensional space,then I can deduce that a strictly convex space with finite dimension is also uniformly ...
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2answers
67 views

what are the difference between metric space and metric linear space?

I know the meaning of metric space and vector space. but i want to differentiate metric space from metric linear space. so basically what are metric space and linear space?
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3answers
100 views

Contraction Mapping Principle

Let $X$ be a Banach space and $T\in\mathscr{L}(X,X)$ with $\|T\|_*<1$. Use the Contraction Mapping Principle to show (where $I$ is the identity map on $X$) that $I-T\in\mathscr{L}(X,X)$ is ...
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1answer
176 views

Fundamental theorem of calculus in Sobolev Space $H^1$

I would like to know whether the the Fundamental theorem of calculus (Part II) can be applied in the following setting. Let $(a,b)$ be an open interval in $R^1$. Let $u \in H^1((a,b))$ with $u(a)=0$ ...
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1answer
32 views

The range of Sobolev spaces to which the function $r^\beta\sin\beta\theta$ belongs

I am learning about Sobolev space, and I am working on the following problem from "The mathematical theory of finite element methods" by Brenner. To make the problem a little bit easier for me, I ...
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1answer
44 views

Bochner Integral: Lebesgue Point

This thread is just a note. Given an euclidean space and a Banach space. Consider Bochner integrable functions: $$F\in\mathcal{B}(\mathbb{R}^d,E):\quad\int\|F\|\mathrm{d}\lambda<\infty$$ Then ...
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2answers
42 views

Exercise on the first and second derivative of a square summable function

Let be $\psi \in L^2[0,2\pi]$, that is the set of the square summable function on the interval $[0,2\pi]$. Then suppose that the second derivative of that function $\psi\;''\in L^2[0,2\pi]$. It is ...
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1answer
57 views

Why is Isom(E,F) open in the set of bounded linear operators between E and F?

Let $ E $ and $ F $ be Banach spaces. According to the lecture notes I'm reading $ Isom(E,F) $ (the set of continuous isomorphisms between $ E $ and $ F $ with continuous inverse) is open in the set ...
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2answers
51 views

Cardinality of the set of all complex sequences converging to zero.

I was asked to show that the set of all complex sequences converging to zero has the same cardinality as the set [0,1]. This is the only hole in a proof that I am working on. I need to show there ...
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1answer
96 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...
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1answer
82 views

Dual space of weighted $L^p(\omega)$

Let $\omega \in A_p$, where $A_p$ is the family of Muckenhoupt weights. I'm wondering what is the topological dual space of $L^p(\omega)$. Is it isometrically isomorphic to $L^q(\omega)$? (1/p + 1/q = ...
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1answer
56 views

Is a weak-star compact subset of $L^\infty$ automatically bounded under the $L^\infty$-norm?

I want to know if a weak-star compact subset of $L^\infty$ automatically bounded under the $L^\infty$-norm. This does not seem to be covered by Arzela-Ascoli, or does it? Thanks.
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1answer
114 views

Approximation of a strongly measurable function by a sequence of simple functions.

Let $(X, \mathcal{A})$ be a measurable space and let $E$ be a normed space. $(i)$ $f:X \rightarrow E$ is called Borel measurable if $f^{-1}(B) \in \mathcal A$ for all $B \in \mathcal{Bo}(E)$ where ...
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0answers
18 views

which space hosts space of probability distributions with weak topology?

Given a separable metric space $A$, let $P(A)$ be the space of probabilities defined on $A$ along with its Borel sigma field. One can define Prohorov metric on $P(A)$, which induces the weak topology. ...
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23 views

Relationship between spectrum of $-\Delta:H^1(M) \to H^{-1}(M)$ and $-\Delta:L^2(0,T;H^1(M)) \to L^2(0,T;H^{-1}(M))$?

Let us take a compact Riemannian manifold $M$. Let us define $-\Delta:H^1(M) \to H^{-1}(M)$ by $$\langle -\Delta u, v \rangle = \int_M \nabla u \nabla v$$ and $-\tilde \Delta:L^2(0,T;H^1(M)) \to ...
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30 views

How can I find first variation

I have a functional: $$F=\int_0^l \left[ V(h)+{1\over 2}\left(h_x\right)^2 \right] dx$$ where $V(h)$ is just a function of $h=h(x,t)$, and $h$ subjects to periodic boundary conditions ...
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1answer
34 views

If $x_j \to x $ and $||Tx_j ||\le k $, show that $||Tx|| \le k $ for a continuous linear operator $T$

Let $T $ be a continuous linear operator. Suppose ${x _j } $ is a sequence in some Banach space $X $, with limit $x $, such that $||Tx _j || \le k $. Show that $||Tx ||\le k $ Well I suppose that I ...
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1answer
40 views

Has a $L^1$ bounded sequence a weak converging subsequence in $L^2$?

Let $f_n \in L^2(0,1)$ with the property that $\sup_n || f_n ||_{L^1}<A< \infty$, i.e. $f_n$ is a sequence in $L^2$ that is uniformly bounded in the $L^1$-Norm. Does $f_n$ then have a weak ...
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2answers
61 views

Banach Spaces which are not $L^p$

Most of the times, when I think of Banach Spaces I think of $L^p$ spaces. I would like to know if there is any Banach space which can not be written as $L^p$ space. Please also indicate any ...
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1answer
50 views

Question on a special Derivative

I have this functional defined from a Hilbert space $H$, $J\colon H\rightarrow \mathbb{R}$ defined by: $$ J(u)=\frac12 \|u\|^2-\int_0^1(A(su),u) ds $$ where $A\colon H\rightarrow H$ is a potential ...
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1answer
115 views

dual of $L^\infty$ or $l^{\infty}$ spaces and characters of these spaces

Let $\nu$ be a $\sigma$-additive probability measure on some standard Borel space $(X,\Sigma)$. By Gelfand's transform or by Stone-Cech compactification $L^{\infty}(X,\nu)$ is isomorphic to ...
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1answer
69 views

Example of maximal monotone operators in non-reflexive Banach spaces with applications in PDE

My question is about examples of maximal monotone operators that are defined in non-reflexive Banach spaces and have applications in PDEs, variational inequalities, etc (any application actually)? If ...
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1answer
44 views

A $\| \cdot \|_2$-closed subspace of $C[0,1]$ is always Banach.

I was doing a problem and I realize that if I prove that given a Y $\|\cdot \|_2$-closed vectorial subspace of $\mathcal{C}[0,1]$ is Banach, I'm done. Well, what I've been trying is establishing a ...
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0answers
54 views

Strict Log-Concavity

Let $F:\mathbb{R} \rightarrow [0,1]$ be a cumulative distribution function on $R$, and suppose that $F$ is a continously differentiable function, with derivative $f > 0$. I would like to prove that ...
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0answers
33 views

Is $W(0,T;H^1, L^2) \cap L^\infty(0,T;L^\infty(M))$ dense in $W(0,T;H^1, H^{-1})$?

Let $M$ be a compact Riemannian manifold that is closed. Define $$W(0,T, H^1, L^2) = \{ u \in L^2(0,T;H^1(M)) \mid u_t \in L^2(0,T;L^2(M))\}$$ $$W(0,T, H^1, H^{-1}) = \{ u \in L^2(0,T;H^1(M)) \mid u_t ...
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2answers
154 views

A question about locally compact Hausdorff space

If $X$ is a locally compact Hausdorff space, $C_{0}(X)$ denotes the set of continuous functions from $X$ to $\mathbb{C}$ vanishes at infinity. This is a basic example in C*algebra. My question is Why ...
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2answers
69 views

Using Lagrange's Mean Value Theorem to prove equality of norms

I'm looking for a proof using Lagrange's Mean Value Theorem of the following: Prove that in $C^1[0,1]$ vector-space, $\left\|f\right\|=\left|f(0)\right|+\left\|f'\right\|_\infty$ norm is equivalent ...