10
votes
Smooth Urysohn's lemma on Fréchet spaces
A bump function on a Banach space $X$ is a non-zero function with bounded support. Existence of a bump function of a given smooth regularity has, of course, strong immediate consequences --by ...
- 55.1k
8
votes
Accepted
Smooth Urysohn's lemma on Fréchet spaces
Warning: Since we will be working outside of Banach spaces, one needs to decide the concept of smoothness applied in the following. Since I am citing from Kriegl/Michor's book 1, the default will be ...
- 2,013
5
votes
An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible
(In the following I assume that the word "invertible" in the question means "bijective".)
Your assumptions do not imply that $F$ is bijective (however, they imply that $F$ is ...
- 11.4k
4
votes
Accepted
Approximating continuous functions from $K\times L$ into $[0,1]$
The following provides a construction of $g_i, h_i$ satisfying the required conditions (the ranges of these functions can even be in $[0, 1]$):
Claim: For each $k \in K$, there exists an open ...
- 1,089
2
votes
When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?
Weak$^*$-convergence of a net means pointwise convergence on singletons, and therefore (uniform) convergence finite sets. Uniform convergence on compact subsets means exactly that. Since finite sets ...
- 41
2
votes
Accepted
Computation of tangent functional
$\newcommand{\om}{\omega}\newcommand{\Om}{\Omega} \newcommand{\de}{\delta}\renewcommand{\th}{\theta}$For each real $t>0$ and some $\om_t\in\Om$,
\begin{equation*}
\|x+ty\|
=|(x+ty)(\om_t)|=|...
- 109k
2
votes
Computation of tangent functional
The idea is that, for $|x(\omega)|\sim 1$ and $t$ small, $$|x(\omega)+ty(\omega)|\sim|x(\omega)^2+tx(\omega)y(\omega)|\sim 1+tx(\omega)y(\omega)$$
Indeed, if $|x(\omega)|<1-\varepsilon$, then for ...
- 354
2
votes
Accepted
Gateaux differentiability of the norm in Banach spaces
We have the following definitions:
\begin{equation*}
V^*:=\{f\in S^*\colon\exists x\in S\ f(x)=1\},
\end{equation*}
where $S$ and $S^*$ are the unit spheres in the underlying Banach space $E$ and ...
- 109k
2
votes
Accepted
Potentially elementary question on affine functions on Banach spaces
A function $f$ from a linear (or, more generally, affine) space $A$ is affine (see e.g. this post) if for all $a$ and $b$ in $A$ and all real $t$ one has $f((1-t)a+tb)=(1-t)f(a)+tf(b)$. If $f$ is ...
- 109k
2
votes
Accepted
Definition and properties of tangent functional
Take any $x\in S$ and $y\in E$. We have to show that
\begin{equation*}
\tau(x,y)\overset{\text{(?)}}=\rho(x,y):=\sup_{f\in T(x)}f(y). \tag{1}\label{1}
\end{equation*}
Generalize the definition of $T(\...
- 109k
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