New answers tagged banach-spaces
7
votes
Accepted
Existence of pairwise quasi-complementary but not complementary subspaces
It is well known that every subspace $Y$ of a separable Banach space $X$ is quasi-complemented.We denote its quasi-complement by $Z$. This is a classical result due to F. J. Murray and G. Mackey.The ...
- 966
4
votes
Accepted
Existence of a complemented basic sequence
No. Google "hereditarily indecomposable Banach spaces" to see that there exist separable Banach spaces in which all complemented subspaces are either of finite dimension or of finite ...
- 31.3k
4
votes
A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
Umm, $A$ is path-connected and it is fairly simple -- just change the coordinates one by one. Say we want to go from $u\in A$ to $v\in A$. On $[0, \frac{1}{2}]$ change linearly $u_1$ to $v_1$, then on ...
- 6,021
2
votes
A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
For the first question, yes, $S \setminus A$ is contractible. Rewrite the standard basis so that it is indexed by $\mathbb{Z}$. Let $U$ be the bilateral shift. Then $U$ is an isomorphism on $\ell^p$ ...
- 2,314
5
votes
Accepted
A more general product rule for weak derivatives?
For $\varepsilon>0$ consider a continuous function $\theta_\varepsilon:\mathbf{R}_{>0}\rightarrow\mathbf{R}_{>0}$ equaling the identity map on $I_\varepsilon:=(\varepsilon,\varepsilon^{-1})$ ...
- 3,325
7
votes
Accepted
Characterization of normed spaces based on violation of parallelogram law
If the above inequality holds for all nonzero $x,y$, then if all of $x,y,x+y,x-y$ are nonzero, we also have (applying your inequality to $x+y$ and $x-y$):
$$
\frac{1}{2} \frac{\|2x\|^2 + \|2y\|^2}{\|x+...
- 10.4k
3
votes
Non-complete space verifying uniform boundedness
Locally convex spaces which satisfy the uniform bounded principle, i.e., every pointwise bounded family of continuous linear maps (with values in any normed space) is equicontinuous, are called ...
- 16.2k
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