Questions tagged [locally-convex-spaces]

For questions about topological vector spaces whose topology is locally convex, that is, there is a basis of neighborhoods of the origin which consists of convex open sets. This tag has to be used with (topological-vector-spaces) and often with (functional-analysis).

7
votes
0answers
261 views

If locally convex topologies exhibit the same dual spaces, do they exhibit the same continuous linear operators?

Consider the following setting: Let $X, Y$ be vector spaces over the field $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$. Furthermore, let $\tau_1, \tau_2$ be locally convex topologies on $X$ and $\...
3
votes
1answer
473 views

How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?

I am reading a theorem about measurability of vector-valued functions in a note on functional analysis: Theorem 3.6.1. If $X$ is a separable, metrizable locally convex space, $(\Omega, \Sigma, \mu)$...
18
votes
1answer
852 views

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to ...
6
votes
6answers
3k views

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 ...
3
votes
1answer
479 views

linear subspace of dual space

$X$ is a locally convex space and $X^*$ is its dual space with weak* topology or uniform topology. If $H$ is a linear subspace of $X^*$ such that $\bar H \ne X^*$, then is there a non-zero $x \in X$ ...
6
votes
1answer
8k views

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 ...
4
votes
1answer
1k views

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 ...
2
votes
1answer
113 views

One assumption in the proof of one result of Hahn-Banach theorem.

Theorem: Let $X$ be a locally convex space, let $M$ be a linear subspace of $X$, and let $f\in M^*$. Then there exists $h\in X^*$ such that $h(x)=f(x)$ for all $x\in M$. I saw one sentence in the ...
1
vote
1answer
89 views

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 ...
0
votes
1answer
351 views

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 ...
7
votes
1answer
267 views

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 ...
1
vote
1answer
1k views

directional derivative of convex function sublinear proving that fact

How can we show that the directional derivative of a proper convex function on $\mathbb{R}^n$ is sublinear? Thank you!
7
votes
2answers
296 views

Is the topology that has the same sequential convergence with a metrizable topology equivalent as that topology?

Let $\mathscr T_1$ and $\mathscr T_2$ be two topologies on space $X$. Assume that $(X,\mathscr T_1)$ is metrizable, and any sequence in $X$ that converges in one of the two topologies must also ...
6
votes
1answer
554 views

Evaluation map is not continuous always.

Let $E$ be a not normable locally convex space, define $$F: E'\times E\to \mathbb R$$ $$(f,e)\to f(e)$$ I have to show that $F$ is not continuous when $E'\times E$ is given product topology. I was ...
8
votes
1answer
258 views

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 $...
7
votes
1answer
553 views

Uniqueness of the derivative in locally convex topological vector space

I need a hint of proof of uniqueness of the derivative in locally convex topological vector space (it's asserted in Lang's "Introduction to differentiable manifolds"). Define derivative of a function ...
6
votes
1answer
683 views

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 ...
6
votes
2answers
888 views

Topology on the space of test functions

I try to read into the theory of distributions and there is one thing which bothers me. I read that a distribution is a linear, continuous functional from the space of test functions, which, depending ...
4
votes
1answer
164 views

Openness of linear mapping 2

I quote a previously asked question : Let $X$ be a topological vector space over the field $K$, where $K=\mathbb{R}$ or $K=\mathbb{C}$, and let $\mathbb\{f\colon X\rightarrow K^n\}$ ($n \in \mathbb{...
3
votes
1answer
294 views

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 \...
6
votes
1answer
212 views

Conceptual question about Locally Convex Spaces

Suppose we have a locally convex space $(V,P)$, where $V$ is a topological vector space and $P$ is a family of seminorms defined on $V$ such that for each nonzero $x\in V$ there is a $p\in P$ such ...
3
votes
2answers
217 views

Closedness in the proof of the Alaoglu Theorem

I'm reading a proof the Bourbaki-Alaoglu Theorem: Could someone explain how the closedness of $\Phi(V^\circ)$ (namely, $\Phi(V^\circ)$ contains all of its limit points) is done in the proof? I ...
3
votes
2answers
222 views

About the convexity of Ky Fan's norm

As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
3
votes
1answer
435 views

Extending continuous, densely-defined linear maps between locally convex spaces

Let $X$ and $Y$ be locally convex topological vector spaces, say over $\mathbb{C}$. To set the stage a bit, I'll say that the topology on $X$ is given by a separating family of semi-norms $(p_i)_{i \...
3
votes
1answer
118 views

Nuclearity of $\mathscr{S}$

I have big problems proving that the Schwartz Space $\mathscr{S}(\mathbb{R}^n)$ together with the topology induced by the family $$ \|\varphi\|_{p}:=\sup_{x\in \mathbb{R^n}}\sup_{|\alpha|\leq p}|(1+|...
2
votes
1answer
364 views

Characterize polytopes resulting from cutting a convex polytope by a hyperplane

We have a convex polytope $P$ for which we know its set of vertices. Using this set we characterize the H-representation of $P$ as: $\mathbf{A}\mathbf{x} < \mathbf{b}$. If a hyperplane defined by ...
2
votes
0answers
1k views

Convex hull of bounded set is bounded

I want to show that in a locally convex topological vector space $X$, the convex hull of a bounded set is bounded. Apparantly this does not hold if $X$ is not locally convex. So the fact that that ...
2
votes
0answers
515 views

Prove that the set of extreme points of a compact convex set is not empty.

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is ...
2
votes
1answer
161 views

Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology ...
1
vote
1answer
179 views

Why sequential continuity from $E$ to $E'$ implies continuity?

Let $(E,\|\cdot\|)$ be a separable Banach space. Let $E'$ be the topological dual of $E$ equipped with the weak* topology $w^*$. I read that a certain linear operator $J:(E,\|\cdot\|)\to (E',w^*)$ is ...
1
vote
1answer
297 views

Locally convex topological vector space

In $C(\mathbb{R})$ space we define two families of seminorms: $p_x(f)=|f(x)|$ where $ x\in\mathbb{Q}$ and $q_x(f)=|f(e^x)|$ where $ x \in \mathbb{R}$ I have to check if above families induce locally ...
1
vote
1answer
50 views

properties of non-extreme points

I'm reading a proof of the following lemma Assume that $K$ is a compact convex set in a Hausdorff locally convex space, and $K$ is metrizable with the induced topology. Then the set $\textrm{ex}(K)$...
1
vote
1answer
94 views

properties of a separable metrizable locally convex space

Let $X$ be a separable, metrizable locally convex space. Suppose $V$ is a neighborhood of $0$ and a barrel (closed, absolutely convex, and absorbing). Show that there exist points $y_n\in X\setminus V$...