All Questions
6
questions
11
votes
1
answer
325
views
Can a norm on polynomials be supermultiplicative?
A norm on a real algebra is supermultiplicative when $\lVert f\cdot g\rVert\geq\lVert f\rVert\cdot\lVert g\rVert$ for all $f$ and $g$ in the algebra.
Is there a supermultiplicative norm on $\mathbb R[...
- 5,726
5
votes
2
answers
256
views
Can a norm on polynomials be "almost multiplicative", even for large degrees?
Definition: A norm on a real algebra is called almost multiplicative if there are positive constants $L$ and $U$ such that, for all $f$ and $g$ in the algebra,
$$L\lVert f\rVert\cdot\lVert g\rVert\;\...
- 5,726
1
vote
1
answer
36
views
Wrong defined/boundless $g \rightarrow\sum_{n = 1}^\infty\frac{g(\frac1n)}{2^n}$
Let's consider:
$$f\colon (C[0,1], \Vert\cdot \Vert_1) \ni g\rightarrow \sum_{n=1}^\infty\frac{g(\frac1n)}{2^n}$$
I'm trying to check if this object is well defined.
where $\Vert f \Vert_1 = \int_0^1|...
- 1,787
1
vote
1
answer
133
views
Finding an upper bound on composite $C^{1}$ functions
The conditions for the problem I am currently tackling is as follows:
Let $T, M > 0$. Define :
$X_{T} := \{ u \in C^{1} ([0,T]) : ||u||_{T} \leq 2M \}$
where $||u||_{T} \ = \ ^{\text{sup}}_{t \in (...
- 467
1
vote
0
answers
99
views
Norm equivalence constants
Take a polynomial $g\in\mathbb{R}[\mathbf{x}]$, in $n$ variables and having some degree $d$, with $g(\mathbf{x})\geq 0$. We define the $p$-norms of $g$ as
$$
\vert \vert g \vert \vert _{p} = \left( \...
- 177
2
votes
1
answer
98
views
Show that $\sum_{k=0}^{N} |P(k)| \leq C(N) \int_{0} ^{1} |P(t)| dt $. [closed]
I’m attending a functional analysis course and I am given to solve this problem as an exercise but I’m a little bit disoriented and I don’t know what tools I can use to get it.
Show that, for each $...
- 251