A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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56 views

Unique continuos linear function given a continuous function from a dense space in X to Y (Y is a Banach Space).

Let $X$ be a normed space, let $Y$ be a Banach Space, let $D\subseteq X$ be a dense linear subspace of $X$ and let $L:D\rightarrow Y$ be a continuous linear function. Then there is a unique continuous ...
5
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2answers
81 views

Is this a correct question?

This is an exam question in functional analysis which for me doesn't make sense the way it is written. I am asking you if you agree with me on the modifications that needs to be done in the question. ...
2
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1answer
190 views

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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1answer
115 views

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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1answer
310 views

Operator Norm of a Linear Transformation

PROBLEM For the linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ equipped with the $l^1$-norm, namely, for $x\in\mathbb{R}^n$, $||x||=\sum_{j=1}^n |x_j|$ and similarly for ...
2
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2answers
129 views

Picard iterations of a matrix

I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one. We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
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1answer
84 views

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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3answers
1k views

Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
4
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1answer
156 views

Normed vector spaces and Banach spaces

Let $X$ be a Banach space with norm $||.||$ and let $S$ be a non-empty subset in $X$. Let $F_b(S,X)$ be the vector space of $F(S,X)$ of all functions $f:S \rightarrow X$ such that $\{||f(s)||:s \in ...
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1answer
172 views

Convergence in normed spaces

I consider a sequence of elements $\{f_n\}_n$ in a normed space $X$ such as $\Vert f_n-f\Vert_X\to 0, n\to\infty$. Let $\{Q_n\}_n$ a complex sequence with $Q_n\to Q$ as $n\to\infty$. Does $Q_nf_n$ ...
3
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2answers
260 views

Banach spaces and quotient space

Let $X$ be a normed vector space, $M$ a closed subspace of $X$ such that $M$ and $X/M$ are Banach spaces. Any hint to prove that $X$ must be a Banach space?
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67 views

Linear functional $\mathscr{L}(E,F)$

Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$. Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question: How to prove ...
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0answers
53 views

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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1answer
77 views

How does $\lim A_n$ being not invertible imply $\sup_n\|A_n^{-1}\|=\infty$?

Consider a sequence of operators $\{A_n\}_{n=1}^{\infty}\subset B(X,Y)$, where $X,Y$ are normed vector spaces and $B(X,Y)$ denotes the space of bounded linear operators from $X$ to $Y$. Assume that ...
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1answer
82 views

Study the equivalence of these norms

I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique. So I define the ...
3
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1answer
124 views

If $U$ is finite dimensional, then operator norm is finite

Let $M:U\to V$ be a linear map between normed vector space $U$ and $V$. We know $U$ is finite dimensional (but don't know about $V$). Define $\|M\| = \sup \{\|Mv\|\;:\;\|v\| = 1\}$. I want to show ...
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2answers
67 views

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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0answers
151 views

weakly convergent sequence in $l^1$ [duplicate]

Prove that every weakly convergent sequence in $l^1$ converges. By Riesz on $L^p$ spaces, every linear functional $L\in (l^1 )^*$, is $L(x) = \langle u,x\rangle$ for some $u\in l^{\infty}$ ...
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3answers
69 views

Determining whether $f(x) = \frac{\sin||x||}{e^{||x||}-1}$ for $x \neq 0$, $f(x) = 1$ for $x = 0$ is continuous at $0$

$f: \mathbb R^m \to \mathbb R$ is defined as $$f(x) = \begin{cases}\dfrac{\sin||x||}{e^{||x||}-1} & \text{if $x \ne 0$} \\ 1 & \text{if $x = 0$.}\end{cases}$$ Note that $x$ is a vector in ...
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1answer
58 views

Prove that for every positive integer $d$ there exists $C(d)>0$ such that

for every polynomial $p(x)$ with degree $\leq d$, $\max\limits_{x\in[0,1]}|p'(x)| \leq C(d)\max\limits_{x\in [0,1]} |p(x)|$. There was also a hint given, that says to "use the compactness of a subset ...
0
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1answer
84 views

Linear operator

Is there a linear bounded (continuous) operator T from $c$ (convergent sequences with sup norm) ONTO $l^1$ (with its usual norm)? If it were so (which seems not), using the open mapping theorem we ...
2
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1answer
171 views

Prove that if $A, B $ are convex , $B$ is closed, $C$ is bounded and $A+C \subset B+C$ then $A\subset B$

Given the sets $A,B,C \in \mathbb{R}^n$ such that: $$A+C \subset B+C$$ Show that if $A,B$ are convex, $B$ is closed and $C$ is bounded then $A\subset B$. I kind of understand the geometrical ...
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0answers
98 views

Which are nontrivial examples of analytical functions on Frechet spaces?

Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
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0answers
74 views

Are these two definitions for dual norm equivalent

Suppose there is a p norm, also is a function $\left\|x\right\|_p$, the dual function at $z$ of it is, $$\sup\limits_x\left\{ z^{T} x-\left\|x\right\|_p\right\}$$ The second is, $$ \sup\limits_x ...
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1answer
109 views

Is $(\ell^1 , \| \cdot \| )$ a Norm space?

Suppose $ x= \{x_n \} \in \ell^ 1$ and $\| x \| = \sup | \sum_{k=1}^n x_k | $, let $ \|x\|_1 = \sum_{n=1}^{\infty} |x_n |$ is a norm for $ \ell^1 $ . Is $(\ell^ 1 , \| \cdot \| )$ a Normed ...
5
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1answer
159 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
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1answer
85 views

Proving $\ell_\infty$ is complete

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), ...
4
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1answer
148 views

Existence of a non zero element in the dual

Let $S$ a vector subspace of a normed vector space $X$ such that $\overline{S} \neq X$. Show that, with the Hahn-Banach Theorem (Geometric Version), that there is $F\in X^{\prime}$ such that ...
2
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1answer
89 views

(p-q)-Lipschitz continuity of linear function

I have the following linear function $f(x,y,z) = ax + by + cz.$ I need to prove that f() is (p-q) Lipschitz continuous where $p=1$ and $q=\infty$. For a given two points $(x_1, y_1, z_1)$ and $(x_0, ...
2
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2answers
150 views

If $x\mapsto \| x\|^2$ is uniformly continuous on $E$, the union of all open balls of radius $r$ contained in $E$ is bounded $\forall r > 0$

A subset $E$ contained in $\mathbb{R}^n$ is such that the function $x \mapsto \left\Vert x\right\Vert^2$ is uniformly continuous on $E$. For $r > 0$, let $E_r$ denote the union of all open balls ...
3
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1answer
54 views

How to show $A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$

How can we show $A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of ...
3
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2answers
233 views

$l_1$ equipped with the sup norm is NOT a Banach Space

Prove that $l_1 = \{ x = (x_k)_{k\in\mathbb{N}}\subset \mathbb{R};\ \sum_{k\in\mathbb{N}}\ |x_k| < +\infty \}$ equipped with the norm $\| x\| = \mathrm{sup}_{k\in\mathbb{N}} |x_k|$ is NOT a Banach ...
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0answers
43 views

Finding an orthornormal basis given a bilinear form

Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
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71 views

Find Lipshitz constant

Let $x, y$ be two vectors in $R^n$. Let $\pi$ be permutation on $\{1, \ldots, n\}$ with uniform distribution. Find Lipshitz constaants of the following functions: $$ f(\pi)=\sum_{i=1}^nx_{\pi(i)}y_i ...
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1answer
63 views

$L_{k}^{1}([0,1])$ is a Banach space

Let $L_{k}^{1}([0,1])$ be the space of all $f\in C^{k-1}([0,1])$ such that $f^{(k-1)}$ is absolutely continuous on $[0,1]$ (and hence $f^{(k)}$ exists a.e. and is in $L^{1}([0,1])$). Show that ...
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3answers
48 views

generalization of a normed space

I study analysis and have a problem: I have a normed space for example $(X,M)$ that is not complete, how can I complete the space $X$ with respect to norm $M$? please help me Thanks
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191 views

Is the image of every open set under a non-zero discontinuous linear function dense in $\mathbb{R}$?

Given a normed space $V$ over $\mathbb{R}$, is it true that the image of every open set of $V$ under a non-zero discontinuous linear function $V\to\mathbb{R}$ is dense in $\mathbb{R}$? I couldnt prove ...
2
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1answer
324 views

Banach spaces and their unit sphere

Let $X$ be a normed vector space. Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete ...
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0answers
71 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. ...
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1answer
95 views

Norms on a vector space over $\mathbb{R}$

In an exercise in the book "Topology and groupoids" the following is asked: Let V be a finite dimensional right vector space over $\mathbb{R}$ ($dim_RV=n)$. Show that any $2$ norms on $V$ are ...
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101 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 ...
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3answers
272 views

If $\{x_n\}$ is a Cauchy sequence in a normed vector space, is $\frac{x_n}{\|x_n\|}$ Cauchy?

Let $\{x_n\}$ a Cauchy sequence in a normed vector space $X$. Is $$y_n = \frac{x_n}{\|x_n\|}$$ another Cauchy sequence in $D = \{x\in X : \|x\| = 1\}$? Remark: The idea is prove that if $D$ is ...
5
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1answer
258 views

Distance between a point and closed set in finite dimensional space

Let $X$ be a linear normed space. I need to prove that $X$ is finite dimensional normed space if and only if for every non empty closed set $C$ contained in $X$ and for every $x$ in $X$ the distance ...
4
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1answer
240 views

If a normed space $X$ is reflexive, show that $X'$ is reflexive.

If a normed space $X$ is reflexive, show that $X'$ is reflexive. Suppose $X$ is reflexive. Then by definition the Canonical mapping $J : X \to X''$ defined by $x \mapsto g_x$ where $g_x(f) = f(x)$ is ...
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1answer
137 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 ...
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1answer
33 views

closed subspace of $\ell_1$ such that sequences of alternating terms cover $\ell_1$

If $X$ is a closed subspace of $\ell_1$ such that every sequence $y=(x_{2n})\in\ell_1$ can be seen as the 'every other term' sequence given by some $x=(x_n)\in X$, why must there be a constant $C$ ...
1
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1answer
169 views

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$. My friends and I have literally been pouring over this problem for days now without success. We've been using Hölder's ...
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2answers
115 views

Norm of a function, Smoothness Penalization

I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
2
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1answer
106 views

finding operator norm $T_N$

How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by $T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer . TIA
1
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1answer
48 views

Does $\|z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$ define a norm on $\mathbb R^2$

Does $\| z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$, with $z=(x,y)\in\mathbb R^2$, define a norm on $\mathbb R^2$?