Tagged Questions

0
votes
1answer
38 views

Convergence in normed spaces

I consider a sequence of elements $\{f_n\}_n$ in a normed space $X$ such as $\Vert f_n-f\Vert_X\to 0, n\to\infty$. Let $\{Q_n\}_n$ a complex sequence with $Q_n\to Q$ as $n\to\infty$. Does $Q_nf_n$ ...
0
votes
2answers
49 views

Find a convergent function in metric space

Let $C[−1, 1]$ be the space of continuous functions equipped with the metric $p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$. Then the sequence of functions $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
1
vote
0answers
45 views

weakly convergent sequence in $l^1$ [duplicate]

Prove that every weakly convergent sequence in $l^1$ converges. By Riesz on $L^p$ spaces, every linear functional $L\in (l^1 )^*$, is $L(x) = \langle u,x\rangle$ for some $u\in l^{\infty}$ ...
1
vote
1answer
60 views

Regarding an isomorphism between a subspace of $\ell^{\infty}$ and $\ell^1$

Let $c_0$ be the subspace of $\ell^\infty$ consisting of sequences that converge to $0$. Show that $c_0$ is a closed subspace of $\ell^{\infty}$ whose dual space is isomorphic to $\ell^1$. Conclude ...
0
votes
1answer
190 views

About Banach Spaces And Absolute Convergence Of Seires

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
2
votes
1answer
63 views

Convergence in norm independent of the choice of the norm.

When I read the proof of the Lie product formula in Reed Simon's book on functional analysis (which essentially reduces to showing $\left\Vert X_n - Y_n\right\Vert\rightarrow 0$ as $n\rightarrow 0$ ...
2
votes
3answers
76 views

Boundedness and pointwise convergence imply weak convergence in $\ell^p$

Let $p\in(1,+\infty)$ and consider the space $\ell^p$ with its usual norm. The following are equivalent: (1) $x_n \rightharpoonup x$ (i.e. $x_n$ weakly converges to $x$); (2) $$\exists ...
2
votes
1answer
120 views

Why is $L^3$ weaker than $L^2$?

Someone told me today that if I can show $\Vert A_n-B_n\Vert_3\to 0$ as $n\to \infty$, then claiming $A=B$ as $n\to \infty$ (where $A$ and $B$ are the respective limits of $A_n$ and $B_n$) is a weaker ...
1
vote
1answer
85 views

convergence in function space

Maybe is a silly question, but for some reason I am confused... If $\mathcal{F}$ is a normed space of real functions and $\displaystyle{ f \in \mathcal{\bar F } }$ then there exists a sequence of ...
4
votes
1answer
93 views

Convergence in $L_\infty$ and $L_1$ even if infinite measure space

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. In the literature, assuming the measure space $X$ has finite measure, if $f_n$ converges to ...
2
votes
3answers
114 views

Cauchy in Norm and Weakly converge Implies Norm convergent

Let $X$ be a normed space and $(x_n)$ is a Cauchy sequence in the norm sense. Also assume the $x_n \rightarrow x_0 $ weakly. Then $x_n \rightarrow x_0 $ in norm. What I did:Take $ \varepsilon ...
1
vote
0answers
249 views

Convergence of $L^p$ norms

Given a measure space $X$ with its measure $\mu$, it can be shown (I'll provide a proof if asked for) that $\displaystyle \forall f \in L^\infty(X,\mu),~\textrm{if } \exists p_0:\forall q \geq p_0, ...
6
votes
1answer
663 views

Does the p-norm converge to the max-norm in some norm

Does the $p$-norm on $\mathbf{R}^n$ converge to the max-norm on $\mathbf{R}^n$ as elements in the space of real valued continuous functions on $\mathbf{R}^n$ endowed with some norm? More precisely, ...
1
vote
1answer
69 views

How do you get pointwise convergence in the context of normed spaces [duplicate]

Possible Duplicate: Norm for pointwise convergence Let $V=C([0,1],\mathbf{R})$ be the vector space of continuous real-valued functions on $[0,1]$. Let $(f_n)$ be a sequence in $V$. Then ...