# Tagged Questions

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### Topology induced by Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
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### Check whether a sequence belongs to an open ball

How to check if the sequence x=( x1 , x2 ,...) where xn =1-(1/n) belongs to the open ball B(0,1) in the normed space l^∞ of all bounded sequences with the norm defined by ...
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### Unit balls in normed spaces.

Assume we talk about the $n$ dimensional vector space over the reals. It is easy to see that for any norm the unit ball is a convex symmetric set. And here is my question : Let $A$ be a bounded , ...
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### Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
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### Prove this is a bounded linear operator and find its operator norm?

I have a map $$A:(C[0,1], || \centerdot ||_\infty) \rightarrow \mathbb R, Ax = x(0) \forall x \in C[0,1]$$ and need to prove it's a bounded linear operator, and find its operator norm. I've tried ...
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### Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
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### Is every normed vector space a metric space?

I was trying to prove that every normed space is a metric space, and the first three proprierties came natural. However, when faced with proving the triangle inequality I had a bit of problems. I ...
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### Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
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### Exercise in Tao Analysis Book

I'm currently studying in the book of Analysis of Terry Tao, amazing book by the way. In one exercise I'm not pretty sure about how can do it (I know that will be almost trivial but I'm stuck in it). ...
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### Proving the normed linear space, $V, ||a-b||$ is a metric space (Symmetry)

The following theorem is given in Metric Spaces by O'Searcoid Theorem: Suppose $V$ is a normed linear space. Then the function $d$ defined on $V \times V$ by $(a,b) \to ||a-b||$ is a metric on $V$ ...
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### Prove that $\lim_{n\rightarrow+\infty}\|\sup_{k>n}u_k-u_n\|=0$

Let $\{u_n\}$ be a Cauchy sequence in the space $(\mathbb R,d)$ with $d(x,y)=\|x-y\|$. Prove that $$\lim_{n\rightarrow+\infty}\|\sup_{k>n}u_k-u_n\|=0.$$ This seems to be obviously however I can not ...
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### Is a normed topological space metrizable?

As stated in the title: If there is a norm on a topological space, then we get a metric induced by the norm. Is this true?
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### A metric space $(\Bbb R,d)$ with $d(x,y)=||x-y||$ is complete!

I would like to receive only the hint, how to prove the statement on the heading. I understand that we have to prove that all Cauchy sequences converges in the space $\Bbb R$, e.g. ...
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### If $X^\ast$ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable

Let $X$ be a Banach space such that $X$* (Dual space of $X$) is separable How can we prove that $S_{X^\ast}$ (Unit sphere of $X$*) is also separable Any hints would be appreciated.
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### Proving that Euclidean space having the infinity metric is a complete metric space (stuck)

I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space. I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
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### Space of finite dimensional subspaces is separable

In Bernard Maurey's paper "A Note on Gowers' Dichotomy Theorem" at the top of the 7th page, the following fact is stated that I'm not able to prove: Let $X$ an infinite dimensional separable normed ...
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### Is the metric induced by a norm ''unique''?

Let $(X,\rVert{\cdot}\lVert)$ be a normed vector space. Clearly there is a canonical way to induce a metric (and so a topology) on $X$ by defining $d(x,y)=\rVert x-y\rVert$. Are there other metrics ...
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### Why $C_{00}$ is not complete with respect to $\sup$ norm?

If $$C_{00}:=\{ x=\{x_n\} \in \mathbb{R^\mathbb{N}}: x_n=0, \forall n>k \text{depending on }x\}$$Can you help me to give such a cauchy sequence in $x$ such that does not converge to $C_{00}$.
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### a question on complete metrizable spaces

There is a claim: Let $Y$ be a complete metrizable space. If $Y$ is bounded, i.e., $d(Y) < \infty$, then $\exists C(X,Y)$ is a complete metrizable space. Why here $Y$ need be bounded? ...
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### Find a convergent function in metric space

Let $C[−1, 1]$ be the space of continuous functions equipped with the metric $p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$. Then the sequence of functions $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
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I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ... 0answers 70 views ### Define metric on set and products Let$X$be set. My question is: if adding point$\ast$to$X$to get set$X \cup \{\ast\}$then on countable product$\prod_{n \in \mathbb N_+} X \cup \{\ast\}$I found it possible to define metric. ... 0answers 93 views ### What are norms used for? These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ... 1answer 129 views ### why must a normed space homeomorphic to a complete metric space be complete? Why must a normed space X homeomorphic to a complete metric space Y be complete? I've solved this in the case where Y is a normed space (considering open balls gives Lipschitz equivalence), but am ... 1answer 90 views ### Pseudonormable Product Spaces I want to prove that a product$\prod_{i\in I}X_i$of topological vector spaces is pseudonormable only if a finite number of the factor spaces are also pseudonormable and the rest have the trivial ... 2answers 157 views ### On$L^p$and$\ell^p$If a continuous and infinitely differentiable function$f(x): \mathbb{R}\to\mathbb{C}$is in$L^p$, is it also true that$f(n),\ n\in \mathbb{Z}$is in$\ell^p$? 1answer 142 views ### Why is$L^3$weaker than$L^2$? Someone told me today that if I can show$\Vert A_n-B_n\Vert_3\to 0$as$n\to \infty$, then claiming$A=B$as$n\to \infty$(where$A$and$B$are the respective limits of$A_n$and$B_n$) is a weaker ... 2answers 404 views ### Show that the discrete metric can not be obtained from$X\neq\{0\}$If$X \neq \{ 0\}$is a vector space. How does one go about showing that the discrete metric on$X$cannot be obtained from any norm on$X$? I know this is because$0$does not lie in$X$, but I am ... 1answer 338 views ### Finding a cauchy sequence that does not converge on M We define the following infinity norm on$\mathbb{R}$as follows $$l_\infty(\mathbb{R}) = \{ (x_i)_{i \in \mathbb{N}} \,\mid\, x_i \in \mathbb{R}, \sup_{i\in\mathbb{N}} \left|x_i\right|<\infty \}$$ ... 1answer 117 views ### Question about norms and coarseness of topology I've been thinking about norms and asked myself the following question: If I have two norms$\|\cdot\|_A$and$\|\cdot\|_B$with$\|\cdot\|_A \leq \|\cdot\|_B$, which topology is coarser, that is, ... 3answers 174 views ### How to show convexity of a ball in metric space? If$(X,\|\cdot\|)$is a normed linear space, then how to show any ball$B(x,r)$is convex? I know that if$x,y\in A\subset V$then$[x,y]\subset A$, where$A$is a convex subset of vector space$V$... 2answers 497 views ### How to show that$\mathbb R^n$with the$1$-norm is not isometric to$\mathbb R^n$with the infinity norm for$n>2$? Could you please give me a hint to prove that$\mathbb{R}^n$with the 1-norm$\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$is not isometric to$\mathbb{R}^n$with the infinity-norm ... 1answer 89 views ### Balls and transformed sets in normed vector spaces Let$T$be a surjective, continuous linear operator between two Banach spaces$E$and$F$. Assume that it is$B_F(y_0,4c)\subset \overline{T(B_E(0,1))}$, where$c>0$,$y_0 \in F$($B$is for ... 1answer 65 views ### Something weaker than the Riesz basis I have some function$f$, real valued and continuous. I formed functions$\{f_{m,k}, k \in \mathbb{Z}, m>0\}$such that that$\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$is dense in ... 1answer 111 views ### Practical implications of a vector space being a topological vector space I have a space$V$and I lately discovered that it's a topological vector space. What are the practical implications of that? 0answers 49 views ### existance of the interpolation space Let$X\subset L_1+L_2$and let$Y$be interpolation space between$L_2$and$L_{\infty}.$Given$U:X\longrightarrow Y$. My question is the following: Is there exists space$Z\subset Y$, such that ... 0answers 84 views ### Consequence of metrizability proof - disregard, the question is an error In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if$X$is a separable Banach space, then the (closed) unit ball,$B_{X^{*}}$of ... 2answers 87 views ### Mean value of convergent series Let us in a normed linear space have a sequence$\{a_i\}_{i=1}^\infty$which converges to some value$b$, how can I show that $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{a_i}{n}=b$$ My idea is to use ... 1answer 572 views ### How to show convergence in a metric space? Suppose that$\{x_n\}→x$where$\{x_n\}$is a sequence in a normed space V and$x ∈ V$. Show that$\forall y ∈ V, \{x_n + y\} → x + y$. 1answer 175 views ### Metric on the unit cube Let$X$be$\mathbb{R}^3$with the sup norm$\|\cdot\|_{\infty}$. Let$Y=\{x\in X: \|x\|_{\infty}=1\}$. For$x,y\in Y,y\neq -x$define$d(x,y)$to be the arc length of the path$$Y\cap \{\lambda ... 0answers 72 views ### Embedding tree metric isometrically into$\ell_\infty$I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ... 1answer 1k views ### Banach space of Lipschitz functions Let$X$be a compact metric space, and$F$the space of all lipschitz functions$X \to \mathbf{C}$. Let$|f|_L$be the least Lipschitz constant. We endow$F$with the norm$||f|| = |f|_L + ...
I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes.... Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
I have this problem: Let $E$ be a normed vector space. $S=\{x\in E : ||x||=1\}$. Show that if $S$ is compact then $\dim E$ is finite. This follows directly from the Riesz's lemma, but in the ...