Articles of locally convex spaces

Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C \left \| x \right \|_X$ for some $C \geq 0$ (general definition of continuous inclusions in normed spaces) […]

Can a strictly increasing convex function $F$ meet a line segment in 3 places, without being linear?

$$(1) F:[0,1] \rightarrow\, [0,1]\,$$ Where a $F$ is a $(C)$ continuous convex function $$\forall t\in [0,1];\forall(x,y)\in[0,1];\,F(tx+(1-t)y)\leq tF(x)+(1-t)\times F(y)$$ satisfying $(E)$ $$(E)\,F(1)\geq 1\,\,, F(0)\leq 0\,\,,\&\, \exists \,\,\text{some third interior fixed point} \,m\in\text{dom}(F);\,F(m)\geq m\,\,\text{where}; 0 \neq m \neq 1$$ Can $F(x)$ meet with a line segment in three distinct points without $F(x)$ being $F(x)=x$? I now know […]

Rainwater theorem, convergence of nets, initial topology

I’ve stumbled upon a result called Rainwater’s theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces. Rainwater’s theorem. Let $X$ be a Banach space, let $\{x_n\}$ be a bounded sequence in $X$ and $x \in X$. If $f(x_n)\to f(x)$ for every $f\in\operatorname{Ext}(B_{X^*})$, then $x_n […]

Krein-Milman theorem and dividing pizza with toppings

In this question the OP mentions the following problem as an exercise on Krein-Milman theorem: You have a great circular pizza with $n$ toppings. Show that you can divide the pizza equitably among $k$ persons, which means every person gets a piece of pizza with exactly $\frac{1}{k}$ of any of the $n$ topping on it. […]

Is $\sqrt{x}$ concave?

I have function $f(x)= \sqrt(x)$. To check is it concave or convex i am checkin $f”(x). $ Which is $ -\frac{1}{4x^{\frac{3}{2}}} < 0$ So the $f(x)$ is concave. Is it correct ? And is is the same with $f(x)=arctan(x)$, $f”(x)= -\frac{2x}{(x^{2}+1)^{2}}<0$. It is concave ?

Reference request: infinite-dimensional manifolds

The following books and/or notes develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of Global Analysis. Choquet-Bruhat & DeWitt-Morette, Analysis, Manifolds and Physics. Klingenberg, Riemannian Geometry. Marsden, Ratiu, and Abraham, Manifolds, Tensor Analysis, and Applications. Hamilton, The inverse function theorem of Nash and […]

Dual space $E'$ is metrizable iff $E$ has a countable basis

I saw that it was already asked, but the book where I’m studying is slightly different. Recall some definition, if $E$ it’s $\mathbb{K}$-vector space and let $\mathcal{E}$ be a vector subspace of the algebraic dual of $E$, wich is the vector space of all linear forms on $E$. We say that $(E,\mathcal{E})$ is a dual […]

Is a sequentially continuous map $f:E'\to E'$ continuous?

I have read that for a separable complete locally convex space $E$, any sequentially continuous linear map $f:E’\to \mathbb{K}$ is continuous (where $E’$ is equipped with the weak*-topology). Is there any similar result for a linear map $f:E’\to E’$?

Convex Hull of Precompact Subset is Precompact

I’m trying to prove that, if $K$ is a precompact (I’ve also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact. I’ve come across a similar statement with Hilbert spaces that suggested I fix some $x\in X$ and define a bounded conjugate linear […]

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please provide any examples? Some thoughts: A complete space is where all Cauchy sequences converge. A normed vector space is a […]