Tagged Questions
2
votes
1answer
53 views
Two inequalities related to norm
We have some difficulties in the following problem:
Let $H$ be a real Hilbert space.
Find $\alpha>0$ such that
$$
\langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq
...
1
vote
1answer
51 views
Orthonormal Family in a Hilbert Space
If we have an orthonormal family, $\{u_n\}_{i=1}^\infty$ in a Hilbert Space $H$, I need to show that for $x\in H$ we have the following inequality:
$$\left|\left\{n|\langle x, u_n \rangle > ...
1
vote
1answer
163 views
Poincaré inequality in unbounded domain
Help me please, how can I to show that Poincaré inequality in unbounded domain doesn't holds?
Thanks a lot!
If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
1
vote
1answer
249 views
Proof by induction of triangle inequality in Hilbert space.
I've made proof by induction over $n$ for triangle inequality : $\left \| x+y \right \|_{e}\leq \left \| x \right \|_{e}+\left \| y \right \|_{e}$
,where $\left \| x \right ...
8
votes
2answers
356 views
Proving an inequality with Cauchy-Schwarz
In the "User's guide to viscosity solutions" by Crandall, Ishii and Lions (link), they make the following claim (inequality (A.4) p. 58) :
Given $x$, $\xi$ $\in \mathbb{R}^n$, $A \in \cal{S}(n)$ ...
9
votes
1answer
305 views
How to prove Halmos’s Inequality
How to prove Halmos’s Inequality?
If $A$ and $B$ are bounded linear operators on a Hilbert space
such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$
I found it from ...
7
votes
2answers
453 views
Contexts For Bessel's Inequality?
Bessel's inequality appears to be about orthonormal sequences.
But (in the context of inner product spaces), I've thought of this inequality as being a demonstration that the hypotenuse of triangles ...
