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This counterexample is so simple so that I might overlook something stupid: Let $T:\ell_1\to c_0$ be the inclusion map. With the natural isomorphisms $\ell_1^*=\ell_\infty$ and $c_0^*=\ell^1$, the tra …

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 barrel …

It seems that a BK-space $X$ should have a continuous linear inclusion (in particular, injective map) into $\mathbb C^{\mathbb N}$ (with the product topology making it a Fréchet space). This is certai …

If $(z_n)_{n\in\mathbb N_0}$ is a weakly convergent sequence with limit $z_0$ then $$ \overline{conv\{z_n:n\in\mathbb N_0\}}= \left\{\sum_{n=0}^\infty\lambda_nz_n: \lambda_n\ge 0, \sum_{n=0}^\infty \l …

Väth only claims that the sequence $y_n$ is Cauchy in the space of measurable functions with the topology (or rather, uniformity) of convergence in measure which, for finite measures, is given by the …

The isometry of $(X\otimes_\pi Y)^*$ and $X^*\otimes_\varepsilon Y^*$ for finite dimensional $Y$ is in 6.1 of the book Tensor Norms and Operator Ideals by Defant and Floret and the isometry of $(X\ot …

I think that question Q3 has nothing to do with Banach space valued functions: If $Y$ is a closed subset of a compact space $X$ such that there is a continuous linear extension operator $F:C(Y)\to C(X …

A colimit is an object $E$ together with morphisms $i_n:E_n\to E$ commuting with the inclusions $i_{n,m}:E_n\to E_m$ (i.e., $i_m\circ i_{n,m}=i_n$) such that, for every sequence of morphisms $f_n:E_n\ …

There is a simple abstract description of the semi-norms of an inductive limit of Banach or locally convex space $E_n$ with linking maps $i_n^m:E_n\to E_m$ for $n\le m$ and $i_n^\infty:E_n\to E_\infty …

(4) follows immediately from a version of the open mapping theorem: If a continuous linear operator between between Banach (or Fréchet or even more general) spaces has non-meager range, then it is ope …

As mentioned in my comment, this is true for reflexive Banach spaces and the compactness game may generalize to other situations, e.g., if the Banach space is a dual space and the embedding in $\sigma …

For the unit ball $B_X$ of the Banach space there are only two possibilities: sep$(B_X)= 1$, if $B_X$ is not separable, and sep$(B_X)=0$ if $B_X$ is separable. Indeed, if sep$(B_X)<1$ there are $\vare …

Since $C(X)$ is not complete one cannot take the closed convex hull of the example in the comment. But what about this: Let $g_n=1-f_n$ with $f_n$ as in my comment. Since the $g_n$ are bounded by one, …

Maybe, I miss something, but the answer seems to be easy: If $f:\mathbb R^n\to\mathbb R^d$ is continuous with $f(0)=0$ you can define the weight function $$\omega(r)=\sup\{\|f(x)\|: \|x\|\le r\}$$ whi …

This is always true (without nuclearity): If $T_j:E_j\to F_j$ are continuous linear maps between Hausdorff locally convex spaces and $E_2$ is complete then $$ T_1\hat\otimes_\varepsilon T_2: E_1 \hat\ …