## 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 ...

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 ...

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 ...

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 $\ell^p$ ...

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 $I_\varepsilon:=(\varepsilon,\varepsilon^{-1})$ ...

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+...

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 ...

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