Linked Questions

2
votes
1answer
981 views

weak convergence of a bounded linear operator [duplicate]

I need help with this problem Let $X$ be a reflexive Banach space and $T: X \to X$ a linear operator. Show that $T$ belongs to $\mathcal{L}(X,X)$ if and only if whenever $\{x_n \}$ converges weakly ...
1
vote
0answers
227 views

Is weakly continuous implies strong continuous? [duplicate]

Let $(X, \|\cdot\|_X), (Y, \|\cdot\|_Y)$ normed spaces, operator $A: X \longrightarrow Y$ weakly continuous. Is $A$ continuous with respect to the strong topology? Thanks in advance for your ideas.
20
votes
1answer
5k views

Weak-to-weak continuous operator which is not norm-continuous

Can one give a "relatively easy" example of a linear mapping $T\colon X\to X$ ($X$ a Banach space) which is a) weak-to-weak continuous b) weak*-to-weak* continuous ($X=Y^*$) but not norm-to-norm ...
9
votes
1answer
2k views

Continuity in weak convergence implies continuity in norm convergence

Let $X$ and $Y$ be norm spaces, and let $T:X\to Y$ be a linear transformation which is continuous under weak convergence. That is, if $\forall x^\star\in X^\star:x^\star x_n\to x^\star x $ then $\...
5
votes
1answer
2k views

If $T: X \to Y$ is norm-norm continuous then it is weak-weak continuous

Let $X,Y$ be normed linear spaces (or Banach spaces if necessary) and let $T: X \to Y$ be linear. We call $T$ norm-norm continuous if $X,Y$ are endowed with the norm topology and similarly, weak-weak ...
1
vote
1answer
517 views

If a linear operator is strong-weak continuous, then it is bounded

$X$ and $Y$ are normed spaces and $L: X\to Y$ is a linear operator from $X$ to $Y$. Show that if $L$ is a continuous operator from $X$ with the strong (norm) topology to $Y$ with the weak topology, so ...
1
vote
1answer
753 views

Weakly sequentially continuous operators in Hilbert space are norm continuous.

Suppose I have a linear operator T from a Hilbert space H to itself, and T maps every weak convergent sequence to a weak convergent sequence. Show that T is continuous. I feel that this statement ...
1
vote
0answers
705 views

Continuous Strong-Strong Implies Continuous Weak-Weak

Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak. I can see that $T$ being ...
3
votes
1answer
62 views

Characterization of boundedness of a linear operator under compactness

The following is an exercise from An Introduction to Banach Space Theory by Robert E. Megginson. Suppose that $T$ is a linear operator from a normed space $X$ into a normed space $Y$. Prove that ...
0
votes
0answers
86 views

Suppose that $ T$ is a linear operator from a normed space $ X $ into normed space $ Y $.Prove that the following are equivalent

Suppose that $T$ is a linear operator from a normed linear space $ X $ into normed linear space $ Y $.Prove that the following are equivalent. $1)$ The operator $T$ is continuous. $2)$The set $ T(K)$...
1
vote
1answer
41 views

A sufficient and necessary condition for the boundedness of linear operator

Let $X$ and $Y$ be two normed linear spaces and $T: X \rightarrow Y$ is a linear operator(mapping). Prove that $T$ is bounded if and only if it maps weakly convergent sequences to weakly convergent ...
1
vote
1answer
33 views

Boundedness of a function on Hilbert space

Given two functions $T,S: H \rightarrow H$, where $H$ is a Hilbert space such that $<Tx,y>= <x, Sy>$ for all $x, y \in H$. How to show that T is a bounded operator? I thought of proving ...