Search Results

Let's recall a simple, elementary, and general fact that hasn't been explicitly mentioned: a dual Banach space is always a splitting subspace in the isometric embedding into its double dual. So $\el …

In general, in order that the intersection be defined, we should assume that $(U,\|\cdot\|_U)$ and $(V,\|\cdot\|_V)$ are continuously embedded into an ambient Hausdorff TVS $E$. Then, $(U\cap V,\|\cdo …

For a counterexample, let $Y$ be a Banach space with a closed, non-complemented subspace $Z\subset Y$. Consider the restriction of the sum operation ${\bf +}:Y\times Y\to Y$ to the space $X:=Y\times …

Let's denote $B$ the unit ball of $X$. Geometrically, we are asking if there is a line $\langle x\rangle$ such that the ball $\alpha B$ can be translated along $\langle x\rangle$ and pass through the …

I understand the aim is an elementary proof; here is one. Let $\omega$ be a modulus of continuity for $f$ on $[0,c]$. Then, for all $0<t\le c-1$ $$\int_0^1|f(x+t)-f(x)|^2dx\le \omega(t)\int_0^1\big(f( …

Let $U\subset \mathbb R^2$ be a bounded open convex neighborhood of $0$. For every $\epsilon>0$ we have $\displaystyle \overline U=\bigcap_{H\supset U\atop H \text{ closed half-plane}} H \subset (1+\e …

This is almost tautological, but suppose that the quasi-nilpotent operator $T$ has the property that, for some $p\in\mathbb N_+$, the equality $\|T^{np}\|=\|T^p\|^n$ holds for infinitely many $n\in\ma …

I would say, monomials are not a Schauder basis for $C[0,1]$ because functions that admits a representation are analytic functions on the unit disc. Actually, $f(t)=\sum_k a_k t^k$ immediately implie …

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 translat …

Say $I:=[a,b]$. We may extend $u$ simply putting it constant for $t\ge b$ and $t\le a$, that is $Eu(t):=u((t\vee a)\wedge b)$. This defines a norm-$1$ linear extensor $E$.

I'd try $X:=\ell_2$ with an equivalent but non strictly convex norm. Let $(e_k)_{k\ge0}$ be the standard Hilbert basis of $\ell_2$. Consider the sets: $A:=\{0\}\cup\{2^{-k}e_k:k\ge 1\}$, $B$, the …

A funny isometry invariant to distinguish these normed spaces is: The space of spheres of radius $2$ in the unit sphere of $(X,\|\cdot\|_X)$ which are maximal by inclusion, as described below. It turn …

The key point is "linear contraction" (which in this context I think means "operator norm not larger that $1$"). Let $(X,\|\;\|)$ be a Banach space and let $\{T_n:E_n\to X\}_{n\in\mathbb N}$ a famil …

Assuming $P$ first countable, the standard contraction principle and elementary bounds are sufficient to conclude. You do not need higher regularity: Let $X$, $Y$ be Banach spaces, $P$ a topological …

Yes, it is true. This inverse function theorem is in a sense half-way between the Lipschitz and the $C^1$ setting. To be more precise let me review some classic results. (Invertibility of Lipschitz p …