# Tagged Questions

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).

59 views
+100

### Continuous semi-norms on subspace

Suppose $X$ is a locally convex topological vector space, let $P$ be the set of all continuous semi-norms on $X$. Suppose $M$ is a subspace of $X$, denote $P|_M$ as the set of semi-norms in $P$ ...
10 views

### Hausdorffness of projective tensor product of locally convex space

I do not see why the following is true: Given two locally convex t.v.s. E and F, if the projective tensor product is Hausdorff then both E and F are Hausdorff. The converse holds and the proof is ...
57 views

18 views

27 views

### Why is $x\in X$ a weak star continuous linear functional in the dual?

I am reading an excerpt from Infinite Dimensional Analysis by Aliprantis and on page 235 it claims that if $X$ is a normed space, then "$x$ is a weak* continuous linear functional by definition". ...
23 views

### Proof of property of unique nearest point equal to convex.

In the picture below, I really don't know how to use the compactness to get there will be a ball $C$ with maximal radius. I report this book for my classmate and teacher. I tell they the $r$ has a ...
12 views

### connect and arcwise-connect in locally convex space

Let X be a locally convex vector space and let G be an open connected subnet of X. How to show that G is arcwise-connected? I only can show that G is path-connected but do not know why G is arcwise-...
44 views

### Do these Topologies define the same open sets?

I am trying to understand weak Topologies by reading John Conway's Course in Functional Analysis and he lists a bunch of theorems such as: "If $X$ is LCS, $(X,wk)^{*} =X^{*}$" which are getting very ...
29 views

### Weak and weak* topologies

If X is a locally convex vectorspace, does the weak and weak* topologies on X* coinside? If so how to prove it?
113 views

### Goldstine theorem and weak topology on $E$ induced by weak* topology on $E''$

In the book where I'm studying there this exercise: "(Goldstine's theorem). Recall that $E \subset E''$, or, more precisely, $J_E(E)$ is a subspace of $E''$ where $J_E :E \longrightarrow E''$ is ...
35 views

### rconvex image under nonlinear function

Let $X\subset R^3$ be a compact and convex set, and let $f: X\rightarrow R^3$ be a nonlinear function, with $f\in C^k$. What are the tools to investigate if the image $K=f(X)$ is also convex, in the ...
42 views

### Open ball around a point in the hyperplane in a locally convex space

Let $f$ be a continuous linear functional in a locally convex space $X$ and let $z\in X$ so that $f(z)=\alpha$. Let $V_z$ be an open neighborhood around $z$. It is my intuition that $V_z$ should ...
15 views

### Existence of Banach space in which nuclear space embeds densely

If $N$ is a nuclear space, does there exist a Banach space $X$, s.t. $N$ embeds densely in $X$?
29 views

### Faithfullness of the Minkowski functional

Let $X$ be a locally convex topological vector space. I need to show that the Minkowski functional $p_C$ for $C$ a convex open neighborhood of $0$ coming from the local base of convex sets, is ...
26 views

### Is the subdifferential always convex and closed set?

Two properties of the subdifferential set are stated as follows: Given a function $f : \mathbb{R}^n → \mathbb{R}$, (i) the subdifferential set $\partial f(x)$ is always convex and closed, even if $f$...
93 views

### Continuity of Minkowski functional in locally convex topological vector space

Let $X$ be a locally convex topological vector space over $\mathbb{R}$ or $\mathbb{C}$ and let $p_C(x)=\inf (\lbrace t>0 \mid t^{-1}x \in C\rbrace)$ be the Minkowski functional for an arbitrary ...
27 views

16 views

It is known that the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ is a Fréchet space and also that the space of test functions $\mathcal{D}(\mathbb{R}^n)$ is dense in $\mathcal{S}(\mathbb{R}^n)$. Let $\... 0answers 37 views ### Show that$f\$(sum of log terms) is convex with Jensen's inequality..
We have the equation $$f(\mathbf{x})=\mathbf{c}^T\mathbf{x}-\sum_{i=1}^m\log{(b_i-\mathbf{a}_i^T\mathbf{x})} ,\;\;\; \mathbf{x} \in \mathbb{R}^n \text{ and } m>n$$ ...