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Motaka
Motaka
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Let $E$ be a Banach space, $T:E\rightarrow E$ ana non-linear operator.

$T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{n} \rightharpoonup x,$ we have $T x_{n} \rightharpoonup T x$.

In papers they use this type of continuity because, "although it is not always possible to show that a given mapping between functional Banach spaces is weakly continuous, quite often its weak sequential continuity can be checked easily. This follows, among other things, from the fact that Lebesgue’s dominated convergence theorem is valid for sequences but it fails for nets".

I have some questions here about strong continuity (in the strong topology) and W.S.C,

  1. In a Banach space, which one implies the other one (if there is an implication between them)? If there are some examples, I will be very grateful
  2. Let's suppose that I am working under the weak topology and I have a result that works for both strong continuity or W.S.C, which one is more interesting to choose?

Let $E$ be a Banach space, $T:E\rightarrow E$ an operator.

$T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{n} \rightharpoonup x,$ we have $T x_{n} \rightharpoonup T x$.

In papers they use this type of continuity because, "although it is not always possible to show that a given mapping between functional Banach spaces is weakly continuous, quite often its weak sequential continuity can be checked easily. This follows, among other things, from the fact that Lebesgue’s dominated convergence theorem is valid for sequences but it fails for nets".

I have some questions here about strong continuity (in the strong topology) and W.S.C,

  1. In a Banach space, which one implies the other one (if there is an implication between them)? If there are some examples, I will be very grateful
  2. Let's suppose that I am working under the weak topology and I have a result that works for both strong continuity or W.S.C, which one is more interesting to choose?

Let $E$ be a Banach space, $T:E\rightarrow E$ a non-linear operator.

$T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{n} \rightharpoonup x,$ we have $T x_{n} \rightharpoonup T x$.

In papers they use this type of continuity because, "although it is not always possible to show that a given mapping between functional Banach spaces is weakly continuous, quite often its weak sequential continuity can be checked easily. This follows, among other things, from the fact that Lebesgue’s dominated convergence theorem is valid for sequences but it fails for nets".

I have some questions here about strong continuity (in the strong topology) and W.S.C,

  1. In a Banach space, which one implies the other one (if there is an implication between them)? If there are some examples, I will be very grateful
  2. Let's suppose that I am working under the weak topology and I have a result that works for both strong continuity or W.S.C, which one is more interesting to choose?
Source Link
Motaka
Motaka
  • 291
  • 2
  • 14

Weak sequential continuity vs strong continuity

Let $E$ be a Banach space, $T:E\rightarrow E$ an operator.

$T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{n} \rightharpoonup x,$ we have $T x_{n} \rightharpoonup T x$.

In papers they use this type of continuity because, "although it is not always possible to show that a given mapping between functional Banach spaces is weakly continuous, quite often its weak sequential continuity can be checked easily. This follows, among other things, from the fact that Lebesgue’s dominated convergence theorem is valid for sequences but it fails for nets".

I have some questions here about strong continuity (in the strong topology) and W.S.C,

  1. In a Banach space, which one implies the other one (if there is an implication between them)? If there are some examples, I will be very grateful
  2. Let's suppose that I am working under the weak topology and I have a result that works for both strong continuity or W.S.C, which one is more interesting to choose?