Tagged Questions
1
vote
1answer
34 views
Proof of non-strictly convexity of $l_1$ and $l_{\infty}$
Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms
$\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty}
...
3
votes
3answers
56 views
Example of two norms on same space, non-equivalent, with one dominating the other
I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
4
votes
1answer
98 views
Linear isometry between $c_0$ and $c$
The following question is an exercise and so I'm just looking for advices and not for answers if it's possible.
I have the following sets in $l^\infty$
$$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
9
votes
1answer
224 views
On the norm of a quotient of a Banach space.
Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm
$$
\left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
0
votes
1answer
23 views
A limit superior question in the context of the Neumann series
I'm trying to understand a step in the proof that the Neumann series converges:
Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
3
votes
0answers
69 views
Absolute norms and 1-unconditional sums
Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, ...
1
vote
1answer
86 views
The Principle of Condensation of Singularities
Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
1
vote
1answer
48 views
How to show that the scalar product on a vector space extends by continuity to a scalar product on the completion of the vector space?
I'm trying to solve the following problem:
Assume $H_0$ is a vector space equipped with a scalar product. Complete $H_0$ with respect to the norm $\Vert x \Vert = \langle x,x \rangle^{1/2}$. We ...
2
votes
1answer
123 views
$C^1 [0,1]$ with different norm
If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where
$$
\Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)|
$$ for any $f\in C^1 [0,1]$, is this space Banach?
...
2
votes
0answers
112 views
Bound on inverse operator
Define
$X = {C^{2, \alpha}}(U \times [0,T])$ and
$Y = {C^{0, \alpha}}(U \times [0,T])$ where $U$ is some real interval.
Let $F:X \to Y$ be a map. Let $DF(g):X \to Y$ be a bounded linear operator for ...
3
votes
2answers
96 views
Geometric Sums in Banach Algebra
Let $E$ be a Banach algebra, and $v\in E$, so that $||v|| < 1$. So the geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm, that part I understand.
I can show that $||w|| \le ...
7
votes
2answers
245 views
Is a completion of an algebraically closed field with respect to a norm also algebraically closed?
Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion ...
