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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39 views

Expressing continuity in terms of seminorms

Let $X$ and $Y$ be locally-convex topological vector spaces, with topologies given by families of seminorms $(p_i) _{i \in I}$ and $(q_j) _{j \in J}$, respectively. If $L : X \to Y$ is a continuous ...
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1answer
49 views

If every real valued continuous function on $A\subseteq \mathbb R^n$ is uniformly continuous , then $A$ is bounded?

Let $A \subseteq \mathbb R^n$ be such that every real valued continuous function on $A$ is uniformly continuous , then $A$ closed and bounded . If $a \in \bar A \setminus A$ , then using the function ...
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54 views

Fixed point problem with matrix

I am looking for guidance for the following fixed point problem. $A$ is an $n\times n$ matrix. The rows of $A=(A_{i})_{i\in N}$ are in the set $A_{i}\in\mathcal{A}$, where $\mathcal{A}$ is the set ...
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1answer
25 views

How are absolute value of a field $F$ and norm of vector space $F/F$ related?

So I think it's true that a field $F$ and its respective vector space $F/F$ are isomorphic, since they consist of the same elements, and the operations of $F$ (addition,mult.) and $F/F$ (vector ...
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41 views

Non-zero bilinear map and uniform continuity

I saw this exercise in a book, but it's not corrected so I'd like your opinion on my solution. Let $E,F,G$ three normed vector spaces and $E\times F$ is equiped with the norm $$||(e,f)||_{E\times F}=\...
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2answers
44 views

Point on the proof that the inverse operator of $I-T$ is given by $(I-T)^{-1}=\sum_{k=0}^\infty T^k$

Let $X$ be a Banach space and let $T\in B(X)$ be such that $\|T\|\lt1$. Suppose then we have the operator $I-T$ and we want to show that its inverse operator $(I-T)^{-1}$ is given by the following ...
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1answer
52 views

A counterexample to $f_{n}\xrightarrow{\|\cdot\|_{L^{1}}}f$ implies $f_{n}\xrightarrow{\|\cdot\|_{C^{0}}}f$

$\|f\|_{L^{1}}\le\|f\|_{C^{0}}$ so $f_{n}\xrightarrow{\|\cdot\|_{C^{0}}}f$ implies $f_{n}\xrightarrow{\|\cdot\|_{L^{1}}}f$. I want to prove the converse is false but cannot come up with a ...
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26 views

Show that the operator $S\in B(Y)$ (the set of bounded linear operators from $Y\to Y$)

I am having trouble with the following problem: "Suppose we have an operator $S:Y\to Y$, where, $$S(g)(y)=g(y)-\int_0^1g(x)dx$$ and where we have $Y=C([0,1])$, equipped with the uniform norm ...
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1answer
43 views

If $\|A\| < 1$, does that imply $A$ is nilpotent?

Suppose $\|A\| < 1$ where $\| \cdot \|$ is the operator norm on matrices, intuitively, $\lim\limits_{k \to \infty} A^k$ goes to zero $\Rightarrow$ $A$ is nilpotent But is this indeed the case? ...
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42 views

Stuck on trivial proof show $(I - A)$ is non singular if $\|A\| < 1$

Assume $A$ is square, $\|A\| < 1$ Attempt: (by contradiction) Suppose $I - A$ is singular, then there exists $x$ such that $x \neq 0$, $\|x\| > 0$, $(I - A)x = 0$ Then $\|(I - A)x\| = 0$ and $...
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0answers
17 views

Prove that the limit of a specific sum of sequences in a normed space is 0

Assume that m sequences $(u_n^k)_{n=1}^\infty$ ($k=1, \ldots, m$) in a normed space satisfy $$\lim_{n\rightarrow\infty} \left(\|\sum _{k=1}^m u_n^k\| - \sum_{k=1}^m \|u_n^k\|\right) = 0$$ Show that ...
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0answers
57 views

p - norms inequality

For $v \in \mathbb{R}^n$ denote by $||v||_p$ its p-norm. That is $(\sum_{i=1}^nv_i^p)^{\frac{1}{p}}$ where $v_i$ are the componenets of $v$. I'm looking for a way to bound the following expression: $$...
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1answer
38 views

Is there only one way to define a norm from an inner product?

Given an inner product $\langle,\rangle$, we can define a norm by $||x|| = \langle x,x \rangle^{\frac{1}{2}}$. My question is, are there other ways to derive a norm from an inner product space and if ...
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1answer
54 views

Proof of inequality in a normed space

Let $E$ be a normed space and let a vector $u\in E$ and a vector $v\in E$ with $||v||=1$ that satisy $||u+v||\geq 2-\varepsilon $ for a $\varepsilon >0$ be given. Show that for all $\alpha ,\beta &...
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1answer
28 views

Will the problem right?

Let $X$ be a normed space and $Y_1$ , $Y_2$ complete spaces and exist two injective linear application $f_1:X \to Y_1$, $f_2:X\to Y_2$ such that $\text{closing } f_1(X)=Y_1$, $\text{closing } f_2(X)...
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1answer
47 views

Can someone explain why $\| \cdot \|_p, p = \frac{1}{2}$ isn't a norm?

$\| \cdot \|_p, p = \frac{1}{2}$ or "half norm" is not a norm What is a quick way to verify that it is indeed not a norm?
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1answer
49 views

Why in normed vector spaces we can define infinite series but in metric space we can not?

We usually define infinite series by partial sums and an inifinite series is said to converge if its partial sum converges. So, if $X$ is a normed vector spaces and $s_n=x_1+...+x_m$ is a partial sum ...
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1answer
175 views

What is a “pseudonorm”?

The following is an excerpt of a note in topological vector spaces. I have tried to search "semi-pseudonorm" on Google but I have got nothing so far. A search with "pseudonorm" returns what we ...
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15 views

Translation of a slice inside the unit ball

Let $B$ the unit closed ball of a $\mathbb R$-normed space $E$, $\ell$ a continuous linear form and $u\in E$ such that $\Vert u\Vert=\Vert \ell\Vert=\ell(u)=1$. Let $H=\lbrace x\in E\,; \ell(x)\ge\...
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1answer
33 views

Quickly sketching the power function $x^{2/3}+y^{2/3}=1$

What is the best way to quickly sketch $x^{2/3} + y^{2/3} = 1$ by hand, without using a graphing device? One can quickly imagine that $x^2 + y^2 = 1$ is a circle. But how does one quickly imagine ...
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1answer
44 views

Absolutely convergence property on normed spaces implying continuity of linear operator

Let $X,Y$ be normed spaces and $f:X\to Y$ a linear operator. Suppose $f$ is such that $\sum_{n=1}^\infty f(x_n)$ is convergent in $Y$ whenever $\sum_{n=1}^\infty \|x_n\| < \infty$. With this ...
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2answers
41 views

An equality in normed space

Let $X$ be a normed space. Suppose that for some $x, y \in X$ we have $\lVert x+y \rVert = \lVert x \rVert + \lVert y \rVert$. Prove that $\lVert \alpha x + \beta y \rVert = \alpha \lVert x \rVert + \...
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2answers
63 views

Why is the norm ball a square in $\,\mathbb R^2\,$ under $\,l^\infty\,$ norm?

Suppose $\,x = \left(x_1, x_2\right)$, then $\,l^2\,$ norm ball is $\,\left\lbrace x\;\big\vert\;\, \sqrt{\left\lvert x_1 \right\rvert^2 + \left\lvert x_2 \right\rvert^2} \leq 1\right\rbrace$ Easily ...
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58 views

Is the norm ball a set or the boundary of a set?

Recall normed ball in $R^2$ under different norms is typically intuited as follows But looking at someone of the definition of normed ball it seems that it describes a closed set rather than the ...
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1answer
17 views

$inf\{||x|| : x \in X , ||F(x)|| = \alpha\} = \frac{\alpha}{||F||}$

If $F \in B(X,Y), F\neq 0$ and $\alpha \geq 0$, then show that $$inf\{||x|| : x \in X , ||F(x)|| = \alpha\} = \frac{\alpha}{||F||}$$ where $B(X,Y)$ is the set of all bounded functions from $X \to Y$ ...
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50 views

Convolution inequality, Gaussian convolution

I'm reading a proof that says for $f\in L^p$ with $p\in[1,\infty)$ we have $\|f\ast p_t-f\|_p\to 0$ as $t\to 0$, where for $t>0$, $p_t(x)=\frac{1}{(2t\pi)^{\frac{d}{2}}} e^{-\frac{\|x\|^2}{2t}}$ is ...
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30 views

Norm of homomorphism embedding Banach algebra into matrix algebra

Let $A$ be a nonunital Banach algebra, and let $\tilde{A}$ be the unitization of $A$, i.e. $\tilde{A}=\{(a,\lambda):a\in A,\lambda\in\mathbb{C}\}$ with multiplication given by $(a,\lambda)(b,\mu)=(ab+\...
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0answers
89 views

Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
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1answer
87 views

Difficulty understanding the proof of equivalence of all norms over $\mathbb R^{n}$

To prove that all norms are equivalent on $\mathbb R^{n}$ , the book I am reading , first takes an arbitrary norm $$|\ \ | \ :\ \mathbb R^{n}\rightarrow\ \mathbb R$$ and then ...
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2answers
105 views

In the proof that $L^{1}$ norm and $L^{2}$ norm are equivalent.

To prove that $L^{1}$ norm , denoted by $||\ \ ||_{1}$ and $L^{2}$ norm , denoted by $||\ \ ||_{2}$ are equivalent we have to find constants $C_{1},\ \ C_{2}$ that satisfies ...
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0answers
35 views

Prove that $\| x \| \le \alpha {\| x \|_{sum}}\,,\,$ $\forall x\in R^n\;$ such that $\alpha=\max_{1\le i\le m}\{\|e_i\|\}\,,\;$ $\|\cdot\|=$ any norm.

Prove that $$\| x \| \le \alpha {\| x \|_{sum}}\quad,\quad \forall x\in \mathbb{R}^n$$ for any norm $\| {\, \cdot \,}\|\quad$ , $\quad{\left\| x \right\|_{sum}}$ is the norm of the sum $\quad$, $\...
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1answer
45 views

Strategy for establishing the triangle inequality of a seminorm

One proof that the $p$-norm $\| x\|_p = (|x_1|^p + \ldots + |x_n|^p)^\frac{1}{p}$ satisfies the triangle inequality exploits the fact that $ x \mapsto |x_1|^p + \ldots + |x_n|^p$ is a convex ...
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0answers
38 views

How to show continuity of $\cdot$ in a normed vector space?

Let $(V,+,\cdot)$ be a normed vector space with the underlying field $K$ . We have to show that $+,\cdot$ are continuous functions. Since $V$ becomes a metric space under this norm so I can use ...
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1answer
57 views

Equivalent norms on $C[0,1]$

For each $f\in C[0,1]$ set $$\|f\|_1 = \left(\int_0^1 |f(x)|^2 dx\right)^{1/2},\quad\quad \|f\|_2 = \left(\int_0^1 (1+x)|f(x)|^2 dx\right)^{1/2}$$ Then prove that $\|\cdot\|_1$ and $\|\cdot\|_2$ are ...
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1answer
34 views

Compute the norm of the operation $A$

Suppose that $\left ( a_{ij} \right )_{i,j=1}^{\infty}$ is a matrix satisfying the following condition $$\sum_{i,j=1}^{\infty} \left | a_{ij} \right |^q < \infty$$ where $q>1$. For $x=\left \{ \...
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1answer
57 views

How strong is the operator norm topology?

Let $(V,\tau_V), (W,\tau_W)$ be normable topological vector spaces. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ inducing $\tau_V, \tau_W$ respectively. Let $||\cdot||_{op}$ be the operator norm ...
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35 views

Norms on $L(V,W)$

Let $V,W$ be normable topological vector spaces over $\mathbb{F}$. Let $C(V,W)$ be the set of continuous linear transformations $T:V\rightarrow W$. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ ...
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1answer
259 views

Operator Norm of a Linear Transformation of a Matrix

The book I am using for the ODE course is Differential Equations and Dynamical Systems by Lawrence Perko. I am having a difficult time understanding what an operator norm of a linear transformation ...
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1answer
27 views

Finding an element in $l_1$ space with certain properties

I am facing a bit problem in the following: Given $x_1,...,x_m \in l^\infty$ and positive $\epsilon_1,...,\epsilon_m$, I need to find an element $a= (a_n)$ in $l_1$ space such that $\sum_{n=1}^ \infty ...
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44 views

Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...
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1answer
77 views

Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
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112 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
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2answers
43 views

The norm of a bounded linear operator has this formula: $\|T\| = \sup_{\|v\| = 1} \|T v\|$

Trying to prove $\|T\| = \sup_{\|v\| = 1} \|T v\|$, given $\|T\| := \inf_{C \geq 0} \{C: \|Tv\| \leq C\|v\|\}$. I know that $\|T(v)\| = \|T(\alpha \hat{v})\| \leq C\|\alpha \hat{v}\|$ for $v = \...
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1answer
227 views

Example of Heine-Borel Theorem

For a subset $S$ of Euclidean space $\mathbb R^n$, S is closed and bounded if and only if S is compact (that is, every open cover of $S$ has a finite subcover). I need an example or ...
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1answer
65 views

If $(V,\|\cdot \|)$ is a finite dimensional space, then all norms are equivalent. [duplicate]

I want to show that if $(V,\|\cdot \|)$ is a finite dimensional space, then all norms are equivalent. I have shown that if $\dim V=m$ all norms $$\|x\|_p=\sqrt[p]{x_1^p+...+x_m^p}$$ are equivalent, ...
1
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3answers
80 views

Equivalence of norms problem.

How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$? I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such ...
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2answers
76 views

What does ||u|| mean?

What does $\left\Vert \mathbf{u}\right\Vert$ mean in this equation? How would this equation be performed? I'm extremely terrible in discrete mathematics and a simplistic answer would be ideal. (Don't ...
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1answer
49 views

What can we say about open unit balls of sup-norm and integral-norm

Consider the normed linear spaces $X_1=(C[0,1], ||.||_1)$ and $X_{\infty}=(C[0,1],||.||_{\infty})$ , where $C[0,1]$ denotes the vector space of all continuous real valued functions on $[0,1]$ and $$||...
0
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2answers
33 views

Can we relax the triangle inequality for $\| v \|$ = $\|v - v_0 + v_0\|$?

Given some vector $v$ on vector space $X$ with a norm $\| \cdot \|$ Then $\| v \|$ = $\|v - v_0 + v_0\|$ where $v_0$ is some other vector is it legal to then write $\| v - v_0 + v_0 \| = \|v -v_0\| +...
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0answers
43 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...