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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4
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1answer
57 views

Confused by peculiar norm

Let $X$ be an infinite subset of $ [0,1]$. In an exercise I am considering the norm on $P([0,1])$ (polynomials on unit interval) defined by: $$||p||_X=\sup_X |p|$$ My question is, how do I make sense ...
0
votes
1answer
158 views

Uncountably many norms such that no two are Lipschitz equivalent

I am struggling with the following question: Is it possible to find uncountably many norms on $C[0,1]$ such that no two are Lipschitz equivalent? I had thought about trying to define norms for each ...
0
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0answers
29 views

Proving that given this property then the norm is induced by a inner product

Let $(X,||\cdot||)$ be a normed space such that, for $x,y\in X$ $$||x + y||^2 +||x - y||^2 = 2||x||^2 + 2||y||^2$$. Then I want to check that $||\cdot||$ is induced by an inner product, so what I ...
2
votes
3answers
64 views

Why aren't the rationals a compact subset of $\mathbb{R}$?

We define a compact subset of some normed vector space $V$ to be any subset $S$ where every sequence $\{\mathbf{x}_{n}\}$ in $S$ has a subsequence which converges to some $\mathbf{x}$ in $S$. Then ...
0
votes
1answer
27 views

If $ T(f)=\int_0^12xf(x)\,dx$ find the value of $||T||$

Let , $T:(C[0,1],||.||_{\infty})\to \mathbb R $ be defined by $\displaystyle T(f)=\int_0^12xf(x)\,dx$ for all $f\in C[0,1]$. Then find $||T||$, where , $\displaystyle ||f||_{\infty}=\sup_{0\le ...
2
votes
2answers
40 views

Separability of $l_p(I,K)$

Good day, I have the following question "Prove that for $1 \leq p < \infty$, $l_p(I,K)$ is separable if and only if $I$ is countable, and $l_{\infty}(I,K)$ is separable if and only if $I$ is ...
0
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1answer
68 views

On continuous mappings on closed unit balls which is injective in the interior

Let $f: B[\theta,1] \to B[\theta , 1]$ be continuous and is injective in $B(\theta , 1)$ ; then is it true that the set $\{x \in Bd \space B(\theta,1): |f^{-1}(\{x\})|\ge3\}$ is countable ? (here ...
0
votes
2answers
31 views

How does one prove that two norms are equal if and only if their closed 1-balls are equal?

Let $X$ be a vector space and let $||\bullet||_1$ and $||\bullet||_2$ be two norms on $X$. I wish to prove that $||x||_1=||x||_2$ for all $x\in X$ if and only if $\{ x\in X \text{ such that ...
1
vote
0answers
36 views

Continuous linear bijection of a Banach space is a homeomorphism

I have seen an example of a continuous linear bijection $f:S\to S$, where $S$ was a normed linear space, such that the inverse function $f^{-1}$ was not continuous,as it was unbounded.The norm on $S$ ...
2
votes
3answers
132 views

$T$ is surjective if and only if the adjoint $T^*$ is an isomorphism (onto its image)

I am trying to prove the following statements: Let $X$ and $Y$ be normed spaces (not necessarily complete) Let $T\in L(X,Y)$ (meaning $T:X\to Y$ is a bounded linear map). Let $T^*:Y^*\to X^*$ denote ...
0
votes
1answer
38 views

Is there any continuous function $f:D^n \to S^{n-1}$ whose restriction to the sphere is the identity?

Is there any continuous function $f:D^n \to S^{n-1}$ whose restriction to the sphere is the identity ? If there does not exist such a function then can we deduce Brouwer fixed point thoerem from this ...
5
votes
0answers
68 views

On the form of injective function $f:\mathbb R^+ \to \mathbb R^+$ such that $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ is a norm

Let $f:\mathbb R^+ \to \mathbb R^+$ be injective such that on $\mathbb R^n , n>1)$ , $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ , where $u=(u_1,u_,\ldots,u_n)$ , defines a norm ; then is it true ...
3
votes
1answer
47 views

Is every closed set , the set of zeroes (resp.critical points) of some smooth real valued function? [duplicate]

Let $A$ be a closed subset of $\mathbb R^n$ : 1) Is it true that for some smooth function $f: \mathbb R^n \to \mathbb R$ , $A=f^{-1}(\{0\})$ 2)Is it true that for some smooth function $f: \mathbb ...
0
votes
0answers
26 views

What does a norm which is symmetric “around” an index subset look like?

I am looking at norms $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ that have the following symmetry property $$\forall \beta \in \mathbb{R}^{n} \text{: }, \lVert\beta \rVert=\lVert ...
1
vote
2answers
56 views

Closed subspace of a functional space and its orthogonal complement

Let's consider the following subspace of $C[-1, 1]$ with the following inner product $(f, g) = \int_{-1}^{1}{f(t) \overline{g(t)} dt}$: $$H_{0} = \{ f \in H | \int_{-1}^{0}{f(t) dt} = ...
0
votes
1answer
28 views

dual pairs in $\mathbb R^n$ with non-standard norms $l_1$ and $l_{\infty}$

Suppose $X=Y=\mathbb R^n$. Usually, to apply a separating hyperplane theorem in $X$ (the primal space), we associate both $X$ and $Y$ with the standard Euclidean norm. My question is that could I ...
1
vote
1answer
26 views

Restrictions of function decomposition in $R^3$

I'm interested in the properties of countable basis functions that span functions living in $\Bbb R^3$. Can I represent a $L^2$ normalizable function that has a point divergence, (for example, ...
1
vote
1answer
41 views

Is it possible to strengthen this inequality?

Let $T:X\to Y$ be a linear operator from a normed space $X$ into a normed space $Y$. Suppose that $T$ has the property that for a fixed $y\in Y$ and any $\alpha>1$, there exists an $x_{\alpha}\in ...
0
votes
1answer
40 views

Composition of a function and a scalar

Are the following true? $\operatorname{scalar} \circ \operatorname{function} = \operatorname{scalar} \times \operatorname{function}$ $\operatorname{function} \circ \operatorname{scalar} = $ the ...
0
votes
1answer
38 views

X is Banach iff $\sum_{n \geq 1} y_n$ converges, where $\left \| y_n \right \| \leq 2^{-n}, \forall n$

Prove that the normed space $X$ is Banach space if and only if $\sum_{n \geq 1} y_n$ converges, where $\left \| y_n \right \| \leq 2^{-n}$ for all $n$.
0
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1answer
50 views

Prove by Induction that Norms in a Finite Dimensional Space are Equivalent?

I would have thought that this is a good candidate for an inductive proof, but I have searched for one and failed. Is there such a proof, and if not why not ? Here's how far I got. It's easy to ...
6
votes
1answer
115 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
17 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
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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
30 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
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
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
88 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
29 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
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 ...
0
votes
1answer
24 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
48 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
53 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
39 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
46 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
34 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
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?
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
64 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
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 ...
2
votes
1answer
55 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
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 ...
6
votes
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 $U$ ...
1
vote
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 ...
0
votes
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 ...
2
votes
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
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
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 ...