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.

learn more… | top users | synonyms

1
vote
1answer
48 views

$|x+y|=|y+x|$ in a normed group

A normed group $(X,+,|\cdot|)$ is a set $X$ equipped with a group operation $+$ and a function $|\cdot|:X\to\Bbb R$ called a norm such that $|x|=0\iff x=0$ $|x-y|\le|x|+|y|$. From ...
2
votes
1answer
26 views

Condition on subsets of normed linear space such that “every real valued continuous function on the subset is uniformly continuous” imply boundedness

If $A$ is a connected subset of a real normed linear space such that every real valued continuous function on $A$ is uniformly continuous , then is it true that $A$ is bounded ? If not , then what if ...
0
votes
1answer
71 views

Want to find the operator norm of a simple matrix, not sure which definition to use

I want to find the operator norm of $A = \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}$ My prof defines the operator norm as $\|A\| = \max_{\|x\| \leq 1} \|Ax\|_2$ In the problem ...
3
votes
1answer
74 views

Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping.

Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping. I think if I can show that $T(B_X)$ contains an ...
1
vote
3answers
82 views

The dual vector space is always complete.

If $N$ is a normed a linear space, then its dual vector space $N^*$ is always complete. Attempt: Let $\{f_n\}$ be a Cauchy sequence in $N^*$. Then, for some $\varepsilon > 0$, there exists $m,n ...
1
vote
1answer
31 views

Every finite dimensional normed linear space has the same dimension as its dual space.

If $N$ is a normed linear space of dimension $n$, then $N^*$ has dimension $n$ as well, where $N^*=L(X,\mathbb{C})$ (all linear functionals from $N$ to $\mathbb{C}$). Attempt: Let $B = ...
1
vote
1answer
35 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 ...
1
vote
1answer
45 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 ...
0
votes
0answers
50 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 ...
0
votes
1answer
24 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 ...
0
votes
0answers
38 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 ...
2
votes
2answers
42 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 ...
1
vote
1answer
51 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 ...
0
votes
0answers
24 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 ...
0
votes
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? ...
0
votes
3answers
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 ...
0
votes
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 ...
1
vote
0answers
51 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: ...
0
votes
1answer
36 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 ...
1
vote
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 ...
0
votes
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 } ...
1
vote
1answer
46 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?
1
vote
1answer
47 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 ...
0
votes
1answer
142 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 ...
0
votes
0answers
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\,; ...
1
vote
1answer
32 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 ...
0
votes
1answer
43 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 ...
1
vote
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 + ...
2
votes
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 ...
0
votes
0answers
51 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 ...
1
vote
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 $X \to Y$ ...
0
votes
0answers
47 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 ...
2
votes
0answers
28 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 ...
1
vote
0answers
80 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 ...
1
vote
1answer
86 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 ...
1
vote
2answers
98 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 ...
0
votes
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 $\quad$, ...
1
vote
1answer
42 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 ...
0
votes
0answers
36 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 ...
0
votes
1answer
52 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 ...
2
votes
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 \{ ...
2
votes
1answer
54 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 ...
0
votes
0answers
34 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 $V,W$ ...
4
votes
1answer
230 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 ...
1
vote
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 ...
0
votes
0answers
43 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 ...
0
votes
1answer
64 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.
1
vote
0answers
98 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 ...
2
votes
2answers
42 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 = ...
0
votes
1answer
190 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 ...