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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6
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1answer
74 views

Reflexive Banach space: Boundedness of subset implies weak compactness. Closed or not?

Claim:In a reflexive Banach space, the weak compactness of a subset is equivalent to the boundedness of the subset. But there is no guarantee that the bounded subset would even have its sequences ...
1
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1answer
16 views

Finding a compact set containing the unit ball in a normed space

I would like to show that there is a compact set $K \supset \{ \Vert x \Vert \leq 1 \}$ in a general normed vector-space $X$, but I have no clue how to do it. Or is it maybe possible to have a finite ...
0
votes
0answers
20 views

Completeness of the space $C_{2\pi} \left ( \mathbb{R} \right )$ [duplicate]

Denote $C_{2\pi} \left ( \mathbb{R} \right )$ is the set of complex-value, continuous on $\mathbb{R}$, periodic functions with period $2\pi$. Fix $p \in \left [1, \infty \right )$. For each $x \in ...
0
votes
0answers
28 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}) $ ...
1
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2answers
25 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
votes
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 ...
2
votes
0answers
69 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 ...
0
votes
1answer
25 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: ...
0
votes
1answer
18 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 ...
0
votes
1answer
23 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( ...
1
vote
2answers
47 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 ...
0
votes
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 ...
0
votes
1answer
45 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 ...
0
votes
2answers
30 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 ...
2
votes
1answer
37 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. ...
0
votes
2answers
33 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 ...
0
votes
1answer
23 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?
1
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3answers
35 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 ...
1
vote
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 ...
0
votes
2answers
48 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 ...
3
votes
2answers
64 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 ...
2
votes
1answer
44 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 ...
0
votes
0answers
15 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 ...
5
votes
1answer
44 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 $U$ ...
1
vote
1answer
39 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 ...
0
votes
0answers
34 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 ...
2
votes
0answers
38 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
votes
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 ...
4
votes
1answer
74 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 ...
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
25 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
44 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
68 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
68 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
24 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
31 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
41 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
49 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
34 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
38 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
23 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
42 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
41 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
45 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
34 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
50 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
27 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 } ...