2
votes
0answers
33 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
3
votes
1answer
30 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
0
votes
1answer
35 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
0
votes
1answer
24 views

weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
1
vote
1answer
35 views

weakly convergent subsequence implies strongly convergent

Statement: Let $X$ be a Banach space If $x_n \rightarrow x$ weakly and every subsequence of $\{x_n\}$ has a strongly convergent subsequence, then $x_n\rightarrow x$ strongly in $X$ Attempt: ?
1
vote
0answers
23 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
2
votes
2answers
45 views

Boundedness of a sequence of functions

Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that $$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ ...
2
votes
1answer
73 views

L1 convergence gives pointwise convergent subsequence

I have been reading Terry Tao's notes on Real Analysis and there's a part he just says, but does not really explain, so I am wondering if someone here would. The notes are ...
1
vote
1answer
38 views

Strong convergence of bounded sequences in Bochner spaces

Let $S=(0,T)$ for a $T>0$ and let $B_0,\ B_1,\ B_2$ be Banach spaces, such that $B_0$ is compactly embedded in $B_1$, which is in turn continuously embedded in $B_2$. Suppose we have a sequence ...
2
votes
1answer
75 views

Convergent sequence in Lp has a subsequence bounded by another Lp function

For $E$ a measurable set and $1\leq p<\infty $, assume $f_n\to f$ in $L^p(E)$. Show that there is a subsequence $\{f_{n_k}\}$ and a function $g\in L^p(E)$ such that $|f_{n_k}|\leq g$ almost ...
5
votes
0answers
52 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
0
votes
1answer
23 views

weak-*-convergence of measures ==> convergence of the total mass?

Let $X = [0,1]$. Let $\mu_n$ be a sequence of regular signed Borel measures on $X$, which converges to a measure $\mu$ on $X$ in weak-star, i.e. for any $f\in C_0(X)$, we have $\int_X f \mu_n(dx) \to ...
0
votes
0answers
11 views

Epi-convergence to indicator function

Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be a continuous approximation of ...
0
votes
1answer
30 views

Why is lower semicontinuity important for epi-convergence?

Why is the lower semicontinuity property important for epi-convergence (and, on the contrary, upper semicontinuity is not desirable)? A simple example would also help.
1
vote
0answers
42 views

Product of strong and weakly converging sequences

Consider a sequence of functions $\{u_n\}\in L^2([0,T],L^2(\Omega)) $ which converges strongly to a function $u\in L^2([0,T],L^2(\Omega))$. Then $u_n \rightarrow u \;\; a.e. $ in ...
2
votes
1answer
46 views

Integral convergence involving Lp and Sobolev spaces

Quick question, any contribution or hint would be appreciated: How does it follow that: $$\lim\limits_{k \rightarrow \infty}\int_{\Omega}a(u_{k})\frac{\partial u_{k}}{\partial x}\frac{\partial ...
3
votes
0answers
78 views

Convergence of sequences of sets

Assume we have the indicator function $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and the sequence of its approximation functions $\{ I_{\nu}(x) ...
3
votes
3answers
46 views

Pointwise Convergence in $L^1$ norm

Suppose we have a sequence of functions $\left\{f_n\right\}_{n=1}^\infty\subseteq C^1([0,1])$ and $f_n\to f\in C([0,1])$ in the $L^1$ norm and $f_n'\to g\in C([0,1])$ in $L^1$. Does it follow that ...
0
votes
2answers
62 views

Is the sequence $(x_n)$ convergent in the space $L_1(0,1)$

Is the sequence $(x_n)$ convergent in the space $L^1(0,1)$ ? $x_n(t)= n^2 t^n (1-t^2)$ for $n\in\mathbb{N}$. norm: $\|x\|=\int_{(0,1)} \left|x(t)\right| \; dt$ I think it should ...
2
votes
1answer
26 views

Totally continuous implies bounded

Consider a separable, reflexive Banach space $V$. We define the mapping $A: V \rightarrow V^{*}$ as totally continuous if it is continuous as a mapping $(V, \text{weak}) \rightarrow (V^{*}, norm)$. I ...
0
votes
1answer
71 views

Find an absotule convergent series that is not convergent

find the sequence of polynomials $(P_n)$ such that $\sum P_n$ converges absolutely (that is $\sum \|P_n\|_{\infty}\lt\infty $) but is not convergent in the space ($\mathcal{P}[0,1], \|.\|_{\infty}$, ...
1
vote
1answer
35 views

Proposed proofs for weak convergence question

I have the following question and two proposed proofs. Please advise if these proofs are adequate and which of the two is better. Thanks. Question: Let $V$ be a reflexive, separable Banach space. ...
0
votes
1answer
14 views

SOT Convergence and Compact Convergence

Let $E$ be a Banach space, and let $A(E)$ denote the closure of the finite rank opertors on $E$. Let $(S_\alpha)$ be a bounded net of operators on $E$ such that $S_\alpha T\to T$ for all $T\in A(E)$. ...
1
vote
1answer
45 views

Showing that a regulated function belongs to Schwartz Space

For clarity I am using the following definition of Schwartz space $\mathscr{S}$ Let $\mathscr{S}$ be the set of functions f(x) s.t. the family of norms ...
1
vote
1answer
35 views

Implications of Weak convergence in Sobolev Spaces

A quick question regarding weak convergence in Sobolev Spaces. If $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$ for bounded $\Omega$ then can we show that $\nabla u_{k} \rightharpoonup \nabla u$ in ...
2
votes
1answer
43 views

Weak Convergence Proof: Tried on My Own

This is related to a question asked here: What a proof of weak convergence is supposed to look like I asked what a proof of weak convergence was "supposed to look like". Specifically, I asked that if ...
0
votes
1answer
31 views

a.e convergence does not imply convergence in $L^{p}$

Do you have an example of $f_n$ which converges a.e but failed to converge in $L^{p}$? Thanks.
0
votes
1answer
39 views

Existence of a sequence of function $f_n$

How to prove that, given a function $f\in L^p$ there exists a function $f_{n}$ compactly supported, in $L^{\infty}$ and such that $f_n \rightarrow f$ in $L^{p}$ ? I think of using the function ...
1
vote
0answers
38 views

Convergence of $L^1$ functions

Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} ...
0
votes
1answer
75 views

Convergence in $L^\infty$ norm and continuous function

Let $\mathcal{C}(T)$ be the set of continuous functions on $T$, which is a metric space under the norm $\left\|f\right\|_{\infty}=\sup_{t\in T}\left|f(t)\right|$. Suppose $\{X_{n}\}$ and $X$ take ...
3
votes
1answer
139 views

Equicontinuity implies uniform convergence

So I know it's a theorem that if $\{f_n\}$ is a sequence in an equicontinuous family of functions defined on a compact metric space $K$ then if for all $x$, $f_n(x)\rightarrow f(x)$ pointwise then ...
2
votes
2answers
45 views

how to prove that the limit of this sequence of functions is continuous?

I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...
1
vote
1answer
31 views

Show that $A$ is non-compact

I have a problem: For $C\left [ 0,1 \right ]=\left \{ x:\left [ 0,1 \right ] \to \Bbb R \ \text{is continuous on } \left [ 0,1 \right ] \right \}$, with a norm: $$\left \| x \right \|=\sup_{t\in ...
1
vote
1answer
57 views

Problem on $L^2$ spaces.

Let $f_n$ be sequence of continuous functions on $[0,1]$ converging uniformly to $f$ a.e. on a set of finite measure. I would like to prove that this implies $f_n\rightarrow f$ in the $L^2$ norm. ...
0
votes
0answers
32 views

Why the convergence of the following operator consequence is strong?

Given a consequence of the following functional operators in $\mathcal{B}(L_{p}[0,1])$, $p \in [1,\infty)$ $$ (A_{n}x)(t) = \sum_{k=1}^{n} n \int_{t_{k-1}}^{t_{k}} x(s)ds \chi_{k,n}(t), $$ where ...
1
vote
1answer
30 views

Relation between convergences in $L^{p}$ for probability spaces.

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...
2
votes
1answer
37 views

Integral convergence and weak convergence

Given that $\Omega \subset \mathbb{R}^{n}$ is a connected bounded Lipshitz domain and $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$. We denote $\Gamma$ as the boundary of the domain. We have the ...
2
votes
1answer
63 views

How Lp spaces relate regarding convergence

I know that for a bounded $\Gamma$ it follows that $L^{q}(\Gamma) \subset L^{p}(\Gamma)$ if $q > p$. I have a few questions regarding how $L^{p}$ spaces relate with regard to convergence. Consider ...
0
votes
2answers
43 views

Better proof for: if $u_n \to u$ in $L^2$ then $F(u_n) \to F(u)$

Let $F(v) = \int_{A} v^2(x)J(x)dx$ where $J$ is bounded. If $u_n \to u$ in $L^2$ then I want to show that $F(u_n) \to F(u)$. The proof is $$F(u_n) - F(u) = \int_A (u_n^2 - u^2)J \leq C\int_A (u_n^2 ...
0
votes
1answer
55 views

absolutely convergent series in Hilbert space

Is it possible to find an infinite dimensional Hilbert space, where every convergent series is absolutely convergent? I could not find any clue to find an example of such type or to disprove that. ...
1
vote
1answer
39 views

Easy question about $C_c^\infty(0,T)$ and $C_c^\infty((0,T);X)$

Let $f \in C_c^\infty(0,T).$ It follows that $f \in C^k(0,T)$ for all $k$, and so if $t_n \to t$ then $$|f(t_n) - f(t)| + |f'(t_n) - f'(t)| + ... +|f^{(k)}(t_n) - f^{(k)}(t)| \to 0$$ for all $k$. Now ...
2
votes
1answer
36 views

Translation is continuous

Let $\mathcal D$ be the space of 'test-functions'. Those are infinitely differentiable functions with compact support. Define the following convergence on $\mathcal D$. $(\phi_j) \to \phi$ in ...
0
votes
0answers
76 views

Absolutely convergent series in normed linear space

I want to prove that in a normed linear space $X$ if for all absolutely convergent series $\sum\limits^{\infty}_{n=1}x_n$, the series $\sum\limits^{\infty}_{n=1}T(x_n)$ is convergent, then $T:X\to Y$ ...
2
votes
1answer
110 views

$x_n$ convergence to $x$ implies $f_n(x_n)$ convergence to $f(x)$. prove that $f$ is continuous

Let $f$ and $f_n$ be functions from $\mathbb{R} \rightarrow \mathbb{R}$ Assume that $f_n (x_n) \rightarrow f (x)$ as $n\rightarrow \infty$ whenever $x_n \rightarrow x$. Prove that $f$ is ...
0
votes
1answer
68 views

Prove that the Set of Bounded Linear Operators is Banach

Let $B(V,V')$ be the vector space formed by set of linear operators $T:V\rightarrow V'$. where $V,V'$ are normed vector spaces. Equip $B(V,V')$ with the norm $$ \|T\|=\sup\frac{\|T(x)\|}{\|x\|} $$ ...
1
vote
0answers
73 views

Topology of pointwise convergence - open sets

Let $X$ be the vector space of all complex functions on $[0,1]$, topologized by the family of seminorms $p_{x}(f)=|f(x)|$, $0\le x\le1$. This topology is called the topology of pointwise convergence. ...
1
vote
1answer
92 views

Prove that the sequence of L-Lipschitz functions converge

$f_n(x): [a,b] \to \mathbb R$ are a sequence of functions that all are $L$-Lipschitz: means - $|f_n(x)-f_n(y)| \le L|x-y|$ , ($L$ is for all the functions) and assume $f_n \to f$ in a pointwise ...
2
votes
1answer
73 views

There exists function sequence $\{f_{n}\}$ converges to $0$ such that $\{a_{n}f_{n}\}$ not converges to $0$

Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$ Show that there exists a ...
2
votes
1answer
30 views

About convergence of $(T_nR_n)$ when $(T_n),(R_n) \subset B(X)$

Let $X$ be a Banach space and $(T_n),(R_n) \subset B(X)$. (a) Prove that if $(T_n)$ converges strongly and $(R_n)$ converges strongly or uniformly, then $(T_nR_n)$ converges strongly (b) Prove that ...
0
votes
1answer
64 views

Constructing Sequences in Lp

Consider Banach Space $L^{p}(U)$ where $U$ is bounded open subset in $\mathbb{R}^{n}$. Take a bounded sequence $\{u_{m}\}_{m}^{\infty}$ in $L^{p}(U)$. Consider a subsequence ...