# Questions tagged [strong-convergence]

A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is an $x\in X$ such that $\lim_{n\rightarrow \infty } \| x_n - x\| =0$

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### Sequence of functions that converges strongly in $L^2(\mathbb R)$ but not pointwise

I am trying to find a sequence of functions $\{f_j\}$ and a function $f$ such that $f_j\to f$ strongly in $L^2(\mathbb R)$ but $f_j$ does not converge to $f$ pointwise. The definition of strong ...
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### Operator satisfying certain properties

Give an example of a Hilbert space $H$ and a sequence of compact operators $(S_n)_{n=1}^{\infty}$ on H such that: (i) $||S_n||\leq 1$ (ii) The operators $V_N=\sum_{n=1}^N\dfrac{1}{n}S_n$ converge ...
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### Poke holes in this proof of the SLLN

I have a proof (sketch) of the Strong Law of Large Numbers, at least the "sufficiency" half of it, that seems a little too easy. This is the version where you only assume i.i.d. random variables, and ...
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### $A$ is strongly dense in its double commutant $A^{''}$

The picture below comes from Murphy's book. My Questions: (1) In the sixth line of the picture, why $u(x)\in K$? Note that from $pu=up$, we only know that $K$ reduces $u$. (2)I think the proof can ...
### Weak convergence and $\|f_{n}\|\rightarrow\|f\|$ implies strong convergence [duplicate]
Let $\left\{ f_{n}\right\} _{n=1}^{\infty}$ be a sequence in $L^{p}\left([0,1]\right)$ for $p\geq1$. Suppose that there exists $f\in L^{p}\left([0,1]\right)$ satisfying \$\lim_{n\rightarrow\infty}...