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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32 views

Invertible linear transformations between a set with infinity norm and euclidean norm

Let $ n \geq 2$. Show that there is no invertible linear transformation between $ S^1 := \{ x:\|x\|_{\infty} = 1\}$ and $S^2 := \{ x : \|x\|_2 =1\} $ as subsets of $( \mathbb{C}^n, \|.\|_{\infty}) $ ...
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2answers
36 views

Norms are not equivalent in $c_0$

Consider two norms in the space $c_0$: $$\lVert x \rVert = \sup \lvert x_i \rvert$$ and $$\lVert x \rVert _0=\sum 2^{-i} \lvert x_i \rvert$$ Prove that above two norms are not equivalent. I know ...
3
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3answers
61 views

What does $\mathbb{R}^\mathbb{N}$ actually mean?

We all know what $\mathbb{R}^n$ means. But I came across this statement about $\mathbb{R}^\mathbb{N}$ in a note that says $\mathbb{R}^\mathbb{N}$ is a vector space under pointwise operations has no ...
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0answers
90 views

Prove that X is a Banach space.

I have this problem to solve and I am not sure if I'm doing it right. Let $X$ denote a real vector space consisting of all continuous functions $u \colon[0, \alpha] \to \mathbb R$ such that their ...
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1answer
30 views

Proof of the continuity method, guidance

Let $\mathcal{B}$ be a Banach space, and $V$ a normed linear space. $L_0,L_1:\mathcal{B}\to V$ are bounded linear operators. Assume $\exists c$ such that $L_t := (1-t)L_0 + tL_1$ satisfies: ...
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1answer
24 views

$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$ Why is $A$ injective?

Why does $A:X\to Y$ where $X$ is a banach space and $Y$ is a normed space, where $A$ is a surjective bounded linear operator, where: $$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$$ Mean that $A$ is also an ...
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1answer
26 views

Should you drop the inner absolute value sign for $L2$ norm?

Lp norm is defined as: $ \left\| \mathbf{x} \right\| _p := \bigg( \sum_{i=1}^n \left| x_i \right| ^p \bigg) ^{1/p}$ But often time I see people writing: $\left\| \mathbf{x} \right\| _2 := \bigg( ...
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2answers
51 views

Duality in finite-dimensional normed spaces

Suppose we endow $\mathbb{R}^n$ with a norm $\|\cdot\|$; call such a normed space $X$. Then, as a vector space, the dual space $X^*$ is also $\mathbb{R}^n$. Let $x\in X$ and $f\in X^*$. Consider the ...
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1answer
25 views

Estimates for $|| \cdot ||_{p}$ and $|| \cdot || _{q}$ norms on $C[a,b]$

Well, i would like to find a minimal constant $C_{a, b, p, q}$ which depends only on $a, b, p, q$ so that the following inequality holds $|| \cdot ||_{p} \leq C || \cdot ||_{q}$, where $1 \leq p \leq ...
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1answer
64 views

An example of an unbounded uniformly continuous function on the open ball of $\ell_2$

It is a consequence of total boundedness of bounded intervals in $\mathbb{R}$ that uniformly continuous functions on such intervals are bounded. What is the best example of an unbounded uniformly ...
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2answers
43 views

Relationship between equivalent norms and ball subsets?

Consider unit balls under norms $\|\cdot\|_i$ and $\|\cdot\|_j$: $$ B(0,1)=\{x\in\mathbb R|\,\,\,\|x\|_i<1 \} $$ $$ \hat B(0,1)=\{x\in\mathbb R|\,\,\,\|x\|_j<1 \} $$ Consider now the ...
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1answer
52 views

Equivalence of definitions of operator norm over general normed vector spaces

A normed module over a general normed ring $(R, |\cdot|)$ is a module with a norm $(V,\|\cdot\|)$ satisfying $\|rx\|=|r|\|x\|$; the norm on the ring is an absolute value in the usual sense, i.e. ...
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36 views

Show that norm is induced by a scalar product

Consider $I = [-1,1]$. Let $C(I)$ be the normed space, equipped with norm \begin{align} ||f||_{2} = \left( \int_{-1}^{1} |f(t)|^2 \, dt \right) ^{1/2} \end{align} Show, that norm is induced by a ...
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1answer
28 views

Examples for permutation invariant norms

I am looking for nice (concrete) examples of permutation invariant norms on $\mathbb{R}^n.$ It is clear that the $\ell_p$ norms do the job. Could you mention me other ones?
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3answers
36 views

Given (X, ||•||) normed space, prove that only X itself and empty space are clopen.

I' d like to ask you for some help. I' ve to prove the problem stated in title, but without using the knowledge that normed space is connected.And I just got no idea how to do so... Thanks for any ...
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1answer
31 views

Find the number of interior points of this subspace of $l^2$.

Consider the Hilbert Space $l^2$. Let $S=\{(x_1,x_2,\cdot\cdot\cdot)\in l^2:\sum\dfrac{x_n}{n}=0\}$. Then find the number of interior points of $S$. Let ...
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2answers
66 views

A norm invariant under permutations but not under signed permutations

Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. We call it axes-symmetric if $\|x\|$ does not depend on the order of the components of $x$. Equivalently if $\|x\| = \|P \cdot x \|$ for any permutation ...
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2answers
66 views

A linear function on the space $c_{00}$ that is not continuous

Consider the space of eventually zero sequences: $$c_{00} = \left\{ x = (x^{(1)},x^{(2)},\dots,x^{(k)},\dots)\in\ell^\infty \,\middle|\, \exists k_0 \text{ such that $x^{(k)}=0$ for ...
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1answer
73 views

Discontinuous bilinear form separately continuous

Do you have an example of a real normed space $V$ and a bilinear form $B : V \times V \to \mathbb R$ that is discontinuous but such that $B$ is separately continuous for each variable? $V$ has to be ...
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0answers
18 views

Proof method for non-equivalence of norms?

Suppose I have 3 norms. I need to prove that any two of them are not equivalent. In my situation, proving that (1 and 3) and (2 and 3) are not equivalent is easy, but proving at (1 and 2) are not ...
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1answer
52 views

A peculiar characterization of open balls in a Banach space

Let $E$ be a Banach space and $U$ be a bounded open subset of $E$. Suppose that for any $x,y\in U$, there exists some open ball $B$ such that $\{x,y\}\subset B\subset U$. Prove that $U$ ...
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1answer
44 views

Does the canonical $\pi: X \to X/Y$ map the closed unit ball to the closed unit ball?

Let $Y \subset X$ be a closed subspace of the normed space $X$. Consider $\pi: X \to X/Y, x \mapsto [x]$. Then for $x \in X, ||x||\le 1$: $\quad||[x]|| = \text{inf}_{y \in Y} ||x-y|| \le ...
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0answers
37 views

If $\lim f'(x) = 0$, then $\lim f(x+1) - f(x) = 0$

Let $f: \mathbb R \to (E,||\cdot||)$, where $(E,||\cdot||)$ is a normed linear space. Suppose that $f$ is differentiable on $\mathbb R$ and that $\lim_{x \to \infty} f'(x) = 0$. Prove that $\lim_{x ...
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0answers
43 views

To show that $p_n(x) \to x$ for every $x \in X$ if and only if $(\|p_n\|)$ is a bounded sequence and $\bigcup_{n=1}^{\infty}R(p_n)$ is dense in $X$

Let $X$ be a Banach Space and Let $(p_n)$ be a sequence of projection operators in $BL(X)$ such that $R(p_n) \subset R(P_{n+1})$ for all $n \in \mathbb{N}$. Then Show that $p_n(x) \to x$ for every ...
5
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1answer
103 views

To show that a matrix defines a map from $l^2$ to $l^2$

Let $$M=\begin{bmatrix} 1 &\frac{1}{2}&\frac{1}{3}&\frac{1}{4} \dots\\ 0 &\frac{1}{2}&\frac{1}{3}&\frac{1}{4} \dots\\ 0 & 0 &\frac{1}{3} &\frac{1}{4} \dots\\ \vdots ...
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1answer
76 views

Continuity of $L_p$ norm in $p$ with $\varepsilon$-$\delta$ definition

Assume that $\|f\|_p< \infty$ for $1\le p<\infty$. In this question we showed that $$ g(p)=\|f\|_p $$ is continuous in $p \ge 1$. The technique was to use Dominant Convergence theorem. Using ...
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1answer
49 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 ...
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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 ...
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1answer
74 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
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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 ...
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3answers
84 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 ...
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1answer
32 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 = ...
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1answer
37 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
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 ...
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0answers
52 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
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 ...
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39 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
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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 ...
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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 ...
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0answers
25 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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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 ...
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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
53 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
37 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
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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 ...
0
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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 } ...
1
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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
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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 ...
0
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1answer
146 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 ...