All Questions
Tagged with hilbert-spaces limits-and-convergence
13 questions
3
votes
1
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266
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Convergence of nuclear operators
Let $H$ be a separable infinite-dimensional real Hilbert space. We consider operators in $H.$
Nuclear norm of a nuclear operator is the sum of its singular values.
A nuclear, positive and self-...
3
votes
2
answers
802
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Spherical harmonics expansion
In the context of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics:
$$
f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{...
- 91
2
votes
0
answers
158
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Question about the ergodic mean
This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question.
I've read a thesis where there is an example on ergodic mean, where however there is ...
- 33
1
vote
1
answer
87
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Convergence in $H^{-2}$ of $L^2$-functions with limit in $L^2$
Assume a sequence $f_n$ in $L^2(\mathbb{R}^d)$ converges in $H^{-2}$ (w.r.t. its norm topology) to a limit $f \in L^2(\mathbb{R}^d)$. In this case, can one improve the convergence, for instance to ...
- 133
1
vote
1
answer
2k
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interchanging limits and summation
So I am stuck at this situation . Suppose $m:B_2(H_1)\times B_2(H_2)\to \mathbb C$ be bilinear form given by $m(S,T)=\left<T,\phi(S)\right>$, where $\phi: B_2(H_1)\to B_2(H_2)$ be a bounded ...
- 243
1
vote
1
answer
496
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Convergence rate of eigenvectors
Let us suppose that $A,A_1,A_2,\ldots$ are non-negative definite self-adjoint bounded linear operators in $L(\mathbb H)$, where $\mathbb H$ is a separable Hilbert space. $(v_j)_{j\ge1}$ and $(\...
- 423
1
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1
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499
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For a bounded sequence in a hilbert space, does $\|u_n\|^2 u_n \to \|u_0\|^2u_0$ ？
If $\{u_n\}$ is bounded in a real Hilbert space $H$, with inner product $(\cdot,\cdot)$, then ${\|u_n\|^2u_n}$ is also bounded.
As there is a weakly converging sub-sequence, we can WLOG assume that $\...
- 11
1
vote
1
answer
73
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If a Hilbert space-valued mapping is norm-decreasing, can we make sense of the limit of sums consisting of projected-values?
Let $H$ be some separable Hilbert space with a given orthonormal basis $\{ e_n \}$. Write the projection onto the subspace spanned by first $N$ basis elements to be $P_N$.\
Now,let $g(t) : [0,1] \to H$...
- 3,477
1
vote
1
answer
81
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Convergence rate of $\operatorname E|\langle X,f_n\rangle|^p$
Suppose that $X$ is a random element with values in a separable Hilbert space $\mathbb H$ such that $\operatorname EX=0$ and $\operatorname E\|X\|^2<\infty$. Suppose that $f_1,f_2,\ldots$ form an ...
- 423
1
vote
0
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Limit of sequence of vectors in $\ell^2$ with coefficients approaching $0$
Let $\{v_m\}_{m \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over the complex plane $\mathbb{C}$ such that: $\{v_m\}_{m \in \mathbb{N}}$ is linearly independend and $v_m \to v$
Let $V= \...
- 433
0
votes
2
answers
1k
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Weak versus strong convergence
This is my first time posting.
I am well aware that an $L^2$ weakly converging sequence is not convergent in the corresponding strong topology. However, my question is as follows, do the sequence of ...
- 213
0
votes
0
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255
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The limit of the operator norm in a Hilbert space
I am not familiar with functional analysis. Could you tell me please, how to prove the following statement (if it is true)?
$$
\lim_\limits{M \to \infty} \|T_A - T_b \| = 0,
$$
here operator norm ...
- 335
-2
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1
answer
1k
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Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]
Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
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