Convergence of Sequences of Operator and Functionals

• Jun 5th 2011, 06:52 PM
geezeela
Convergence of Sequences of Operator and Functionals
Let $\displaystyle {T}_{n}\to T$, where $\displaystyle {T}_{n}\in B(X,Y).$. show that for every $\displaystyle \epsilon > 0$ and every closed ball $\displaystyle K\subset X$ there is an N such that [tex] $\displaystyle \left\|T_nx-Tx\right\| < \epsilon$ for all n > N and $\displaystyle x\in K$
• Jun 5th 2011, 07:16 PM
Drexel28
Quote:

Originally Posted by geezeela
Let $\displaystyle {T}_{n}\to T$, where $\displaystyle {T}_{n}\in B(X,Y).$. show that for every $\displaystyle \epsilon > 0$ and every closed ball $\displaystyle K\subset X$ there is an N such that [tex] $\displaystyle \left\|T_nx-Tx\right\| < \epsilon$ for all n > N and $\displaystyle x\in K$

I just did this somewhere. Where's the effort man? How about using the fact that $\displaystyle \|T_n(x)-T(x)\|\leqslant \|T_n-T\|_\text{op}\|x\|$.
• Jun 5th 2011, 07:41 PM
geezeela
Quote:

Originally Posted by Drexel28
I just did this somewhere. Where's the effort man? How about using the fact that $\displaystyle \|T_n(x)-T(x)\|\leqslant \|T_n-T\|_\text{op}\|x\|$.

what does the {op} stands for?..
• Jun 5th 2011, 07:42 PM
geezeela
Quote:

Originally Posted by Drexel28
I just did this somewhere. Where's the effort man? How about using the fact that $\displaystyle \|T_n(x)-T(x)\|\leqslant \|T_n-T\|_\text{op}\|x\|$.

What does that op stands for??
• Jun 5th 2011, 07:43 PM
Drexel28
Quote:

Originally Posted by geezeela
What does that op stands for??

Operator norm? Isn't that the norm you're putting on $\displaystyle \mathcal{B}(X,Y)$.