Intereting Posts

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Structure sheaf consists of noetherian rings
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What is an example of pairwise independent random variables which are not independent?
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What does limit notation with an underline or an overline mean?
Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$
Is totally disconnected space, Hausdorff?
Why is the cardinality of irrational numbers greater than rational numbers?
Find all positive integers $n$ for which $1 + 5a_n.a_{n + 1}$ is a perfect square.
Geometric justification for the prime spectrum and “generic points”
How do I show that a holomorphic function which satisfies this bound on reciprocals of integers is identically zero?
What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?
Compute $\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space$

A homework problem:

Let $H$ be a Hilbert space.

Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$).Show that $S$ is bounded.

- Dual space of $H^1$
- The reflexivity of the product $L^p(I)\times L^p(I)$
- Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”
- $L_{p}$ distance between a function and its translation
- Bounded operators with prescribed range
- Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

**My attempt**: I’d like to use the closed graph theorem. I take $(x_n)\subset H$ and assume $x_n \rightarrow x$ and $Tx_n\rightarrow y$. I’d like to show $Tx=y$.

So I calculate:

$\|Tx_n-Tx\|^2=\|T(x_n-x)\|^2=|\langle T(x_n-x), T(x_n-x)\rangle|=$

$|\langle x_n-x, T(T(x_n-x))\rangle|\leq \|x_n-x\|\cdot \|T(T(x_n-x))\|$.

So, it’s enough to show that $\|T(T(x_n-x))\|$ is bounded. The fact that $T(x_n)$ converges tells me that $\|T(x_n-x)\|$ is bounded, but I don’t know what about $\|T(T(x_n-x))\|$.

- Representation of a linear functional Lipschitz in total variation
- Do there exist two singular measures whose convolution is absolutely continuous?
- Counterexample for the solvability of $-\Delta u = f$ for $f\in C^2$
- Why is there no space whose dual is $C_\mathbb{R}$?
- Transpose of Volterra operator
- Approximate spectral decomposition
- Inequality regarding norms and weak-star convergence
- Characterisation of one-dimensional Sobolev space
- Definition of Equivalent Norms
- Cauchy-Schwarz Inequality

Assume $x=0$ (which is possible by linearity, working if necessary with $x’_n:=x_n-x$, so $Tx’_n\to Tx_n-Tx=y-Tx$), and write

$$\langle y,y\rangle=\lim_{n\to +\infty}\langle Tx_n,y\rangle\underset{\color{red}{\mbox{sym}}}{=}\lim_{n\to +\infty}\langle x_n,Ty\rangle,$$

and conclude using the fact that $x_n\to 0$.

(so actually, in $\lVert y-Tx_n\rVert$, we just need to take the limit *at one side* of the inner product, not both).

- Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?
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- Example of non-trivial number field
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- A generalization of Kirkman's schoolgirl problem
- How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?
- Number of all labeled, unordered rooted trees with $n$ vertices and $k$ leaves.
- Solving a Nonlinear Recursion
- An outer measure is countable-additive on the measureable sets
- Why if $n \mid m$, then $(a^n-1) \mid (a^m-1)$?