# Tagged Questions

The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.

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### In the Krein-Milman theorem, can the weak closure of the convex hull be replaced by norm-closure?

I have a question on the following formulation of the Krein-Milman theorem: Consider a vector space $X$ equipped with the weak topology induced by a separating space $X^*$ of functionals on $X$. ...
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### from finite to $\sigma$-finite measure space [duplicate]

This might be rather elementary. I have put it at MSE for a while without getting any answers. Here is the question: In the proof of the following theorem, would anyone explain how the general case (...
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### Vector space of all Lebesgue measurable functions

Let $L_0$ be the vector space of all Lebesgue measurable functions on $[0,1]$ with metric $d(f,g)$ = $\int_{0}^{1} |f(t)-g(t)|/( 1+ |f(t)-g(t)| ) dt$ . Show that $d(f_n,f) \to 0$ iff $f_n \to f$ in ...
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### what is a cofinal sequence?

I understand that the subset $\Phi'$ of $\Phi$ is cofinal by looking at Wikipedia https://en.wikipedia.org/wiki/Cofinal_(mathematics) Would anybody explain what the cofinal sequence $(Y_n)$ means?
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### Kadets-Pelczynki criterion to prove Eberlein Smulian Theorem.

I am studying the proof of the Eberlein-Smulian Theorem via basic sequences in the book "Topics in Banach Space Theory" from F.Albiac and N. Kalton. Although, I found myself stuck in the following ...
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### Let $H$ be a real Hilbert space (an euclidian vector space over the real numbers field).

Let $H$ be a real Hilbert space (an euclidian vector space over the real numbers field). If $\{x_n\}$ weakly converges to $x$ and $\|x_n\|\rightarrow \|x\|$, show that $\|x_n-x\|\rightarrow 0$. My ...
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### How are the assumptions used in the proof of Bourbaki-Alaoglu Theorem?

This is a follow up question to a previous one. In the proof of the following theorem, where are the assumptions "Hausdorff" and "locally convex" used?
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### Let $E$ be a topological linear space, if $U$ is a neighbourhood of $0$ why is $U+x$ a neighbourhood of $x$?

Let $E$ be a topological linear space, if $U$ is a neighborhood of $0$ why is $U+x$ a neighborhood of $x$? With linear space I mean a vector space over the real or complex numbers. I know the ...
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### Linear finite-dimensional topological vector space is closed.

Suppose $X$ is a topological vector space and let $Y \subset X$ be its subspace with $\dim Y < \infty$. The goal is to prove that $Y$ is closed in $X$. I know how to prove this fact when $X$ is ...
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### 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 ...
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### Uniqueness of separated (Hausdorff) topology on a $n$-vector space

I recently asked a question on a french forum about the proof that any norms on a finite dimensional (real or complex) vector space are equivalent. Someone showed me the proof of a more powerful ...
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### neighborhood base for the Mackey topology

I'm reading the proof of a theorem due to Mackey in a note of functional analysis: I don't see why the first sentence is clear. By the definition of neighborhood base and locally convex topology, ...
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### Show that $df_x: TM_x\rightarrow TN_{f(x)}$ is well-defined

Let $f\colon M\rightarrow N$ be a $C^\infty$ function between $C^\infty$ manifolds. Show that $$df_x: TM_x\rightarrow TN_{f(x)}$$ defined by $$df_x([c_0]) := [f \circ c_0]$$ is a well-defined map. ...
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### Natural isomorphism between linear space to bilinear space

Let $V$ and $W$ be (not necessarily finite-dimensional) vector spaces. Show that there is a natural isomorphism (meaning an isomorphism that can be described without reference to a basis) between the ...
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### Prove that continuity $\Leftrightarrow$ component continuity

Theorem: Let $\mathbf{f}: S \to \mathbb{R}^n$, where $S \subseteq \mathbb{R}^n$. $\mathbf{f}$ is continuous if and only if its components $\mathit{f}_k, k = 0, \dots, n-1$ are continuous. ...
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### linear independence in a dual pair and its consequence

This is a follow-up question to a previous one: linear independence in a dual pair. The following is from the Topological Vector Spaces by Schaefer: The corollary has been proven independently ...
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### $P(D)f=g$ has a solution $f\in C^\infty(\Omega, E)$ for every $g\in C^\infty(\Omega, E)$?

I read in a paper the following statement: Let $P(D)$ be an elliptic linear partial differential operator with constant coefficients, $\Omega\subset \mathbb R^n$ an open subset and $E$ a Fréchet ...
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I am trying to visualize the basic neighborhoods of the form $V(x_0;\varepsilon,f_1,...,f_n) = \bigcap_{j=1}^n \{ x \in E : |f_j(x-x_0)|<\varepsilon \}$ where $x_0 \in E$, $\varepsilon>0$ and $... 0answers 16 views ### cosine similarity among multiple sets my background is not mathematics, so I will try to do my best to describe. I have multiple sets of elements A0 = {... }, A1 ={..}, .., An ={} I want to: 1. compute a similarity (here cosine for ... 1answer 18 views ### Cardinal functions on TVS Its a well know fact that, in the class of metric spaces, the cardinal invariants density and weight agree. I would like to know if there is a example of a topological vector space$X$for whose ... 3answers 83 views ### The existence of a limit point of a closed set Walter Rudin define a closed set as: 2.18 (d) A subset E of a metric space is closed if every limit point of E is a point of E. I don't see in this definition nothing about the existence of a ... 1answer 38 views ### equivalent definitions of weak topology on a topological vector space Let$(X,\tau)$be a topological vector space over$\Bbb{R}$and$X^*$be the topological dual of$X$. There are two ways to define the weak topology$\tau_w$on$X$:$\tau_w$is the initial topology ... 2answers 95 views ### Is every convergent sequence in a topological vector space bounded? It is known that in a metric space, every convergent sequence is bounded. In a topological vector space (TVS), one also has the notion of bounded sets. Here is my question: Is every convergent ... 0answers 55 views ### Counterexamples of topological vector space How can I prove that an vector space with infinitely many elements endowed with the cofinite topology is not topological vector space? I have an indirect proof: Such an infinite space with the ... 1answer 35 views ### continuity of linear operators between topological vector spaces Let$X$and$Y$be topological vector spaces and$T:X\to Y$a linear operator. If$T$is continuous at$0\in X$then$T$is continuous. Suppose$x_0\in X$and let$U$be a nbhd of$T(x_0)$in$Y$.... 0answers 91 views ### Are Lusin and Souslin spaces sequential or even Fréchet-Urysohn? First some definitions: A Polish space is a separable and completely metrizable topological space. A Hausdorff space is Lusin if it is the image of a Polish space under a bijective continuous map. A ... 0answers 48 views ### If$A$is a balanced and convex set, then$\lambda A + \mu A = (|\lambda| + |\mu|)A$? I have proven that$\lambda A + \mu A \subset (|\lambda| + |\mu|)A$. How do I prove the reverse inclusion? 1answer 53 views ### How can I prove this is a metric? While proving the Banach-Alaoglu theorem, one needs to prove the topology induced by a countable family of seminorms$\rho_n$on a vector space$X$is metrizable if$X$is Hausdorff with that topology.... 1answer 38 views ### How do I prove that if$A$is a balanced subset of a vector space$E$then$\lambda A = |\lambda| A$? I have to somehow use$\left|\frac{\lambda}{|\lambda|}\right|=1=\left|\frac{|\lambda|}{\lambda}\right|$, I don't know how, however. 3answers 48 views ### Is every function from a subset of$[0,\infty)$, which is closed under addition and contains$0$, into a Polish space continuous? Let$I\subseteq[0,\infty)$be closed under addition with$0\in I$and$(E,d)$be a Polish space. Is every function$f:I\to E$continuous? The answer seems to be "yes", if$I\subseteq\mathbb N_0$, ... 0answers 28 views ### Is it possible to compute the volume of a cone on a inner product space? This is a matter of curiosity for me. Volumes are often compute using triple integration. But is it possible to compute volumes on a vector space with an inner product defined on that vector space? 1answer 85 views ### characterization of topological vector spaces It is known that if$X$is a topological vector space (TVS), then all the translations and nontrivial scalar multiplications are homeomorphisms. I'm curious about the following question about which ... 1answer 87 views ### Linear span of general set in topological linear spaces I'm studying Functional Analysis and I'm in doubt with the definition of linear span. The book states that: Let$\mathscr{L}$be a topological linear space and let$\mathscr{M}$be a linear ... 0answers 36 views ### “Algebraic isomorphism$\implies$Homeomorphism” in the topological vector space context This question just pop up when I was trying to solve another problem. Let$X,Y$be vector topological spaces over the field$\mathbb{K}$( with$\mathbb{K}=\mathbb{R}$or$\mathbb{C}$) and let$f:X\...
Let $S \subset X, Y$ be normed spaces over $K$. An operator $A:S \to Y$ is called compact if: $A$ is continuous $A$ transforms bounded set into relatively compact sets i.e. if $(c_n)$ is ...
### Is $C^{\infty}(\mathbb{T})$ dense in $C(\mathbb{T})$?
For a topological space $X$, the space of smooth functions with compact support (denoted by $C^{\infty}_0(X)$) is dense in the space of continuous functions vanishing at infinity (denoted \$C_{\infty}(...