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

0
votes
0answers
3 views

Behavior of norm on matrix algebra under multiplying matrix of scalars

If $A$ is a Banach algebra and I equip $M_n(A)$ with the norm $\|[a_{ij}]\| = \max_i\sum_j \|a_{ij}\|$, do I have $\|Z[a_{ij}]\|\leq\|Z\| \|[a_{ij}]\|$ when $Z$ is a matrix with scalar entries and ...
1
vote
2answers
24 views

Prove that the normed spaces $(C[0,1], \| \cdot\|_2)$ and $(C[a,b], \|\cdot \|_2)$ where $\| \cdot\|_2$ is the Euclidean norm are isometric.

Essentially I'm looking for a bijection $f: C[0,1]\to C[a,b]$ such that$$\|f(x) \|_2=\| x\|_2$$ I don't know how to go about finding this function, but I do know that it is possible. $$\| x \|_2 = ...
0
votes
1answer
26 views

Define a metric using scalar product and prove that it is indeed a metric

So this is how I went about this: $\langle\,\cdot\,,\,\cdot\,\rangle: X \times X \to \mathbb{R}$ such that (by definition I list the properties of scalar product) and I can east prove the first three ...
4
votes
1answer
12 views

Proof that every polilinear map who's domain is $R^{n_1} \times R^{n_2}… \times R^{n_k}$ and co-domain any given real normed space Y is bound.

A Polilinear map\operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
1
vote
1answer
33 views

Can anyone help out with this proof, certain steps are unclear. Norm of linear operator.

I have the following norm defined as follows (in $R^n$, $x=(x^1,x^2,\ldots,x^n)\ $)$\| x\|_1= \sum_{i=1}^{n}|x^i|$ Let $A:R^m \to R^n$ a linear map of the spaces $(R^m ,\| \cdot \|_1 )$ and $((R^n ...
1
vote
1answer
14 views

Comparison of the norms of two non-negative real-valued vectors differing only in one component.

Let $0 \leq a \leq b$ and let $\mathbf{x} \in \mathbb{R^{n}}$. Let $\|.\|$ be a norm over $\mathbb{R}^{n+1}$. If we write $( \mathbf{x},a)$ the vector of $ \mathbb{R}^{n+1}$ made by concatening ...
2
votes
0answers
24 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
-2
votes
0answers
25 views

normed spaces and their completion [on hold]

Let $V$ a normed space and $(W,i)$ their completion. Let $U$ a Banach space and $T:V\rightarrow U$ a continuous linear operator. $(a)$ Prove that exists an unique continuous linear operator $ ...
0
votes
0answers
11 views

Bounded v.s. completely bounded homomorphisms between $L^p$ operator algebras

Take an $L^p$ operator algebra to mean a closed subalgebra $A\subset B(L^p(X,\mu))$ for some ("nice") measure space $(X,\mu)$, $p\in[1,\infty)$. Equip the matrix algebra $M_n(A)$ with the norm ...
0
votes
1answer
49 views

prove the continuity of $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$ [duplicate]

Let $\phi\in C[0,1]$ and let $T_\phi~:C[0,1]\rightarrow\mathbb{R}$, defined as $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$. How can i prove that it's a continuous operator?
1
vote
1answer
32 views

Prove that $Hom(V,W)\neq L(V,W)$

Let $V$ a normed space of infinite dimension and let $W\neq 0$ a normed space. Prove that $Hom(V,W)\neq L(V,W)$.
0
votes
2answers
47 views

Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous

Let $k:[0,1]\times[0,1]\rightarrow \mathbb{R}$ continuous and $K:C[0,1]\rightarrow C[0,1]$, given by $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$. Prove that $K$ is continuous. I try to see continuity in $0$, ...
3
votes
2answers
37 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...
-2
votes
1answer
42 views

L(V,W) is Banach, then W is Banach [duplicate]

Let $V,W$ normed vector spaces, $V$ not empty and with a finite dimension. Prove that $L(V,W)$ is Banach, then $W$ is also Banach.
0
votes
2answers
26 views

linear operator on normed spaces

Let $V$ and $W$, normed spaces and $T:V \to W$ a linear operator. How to prove that: "if $T$ is continuous in $0$, so, $\forall A \subset V$ bounded, $T(A)$ is also bounded"
7
votes
1answer
53 views

Topological space $\nRightarrow$ Metric space $\nRightarrow$ Normed space $\nRightarrow$ Inner product space (Examples)

If I have an inner product space, the hierarchy goes: Inner product space $\Rightarrow$ normed space $\Rightarrow$ metric space $\Rightarrow$ topological space. The reverse, however, is not always ...
0
votes
1answer
49 views

how shall i'll prove if c={(x_n) :exists lim x_n}is a Hyperplane, dense, or closed? [on hold]

Let $$c=\{(x_n) :\exists ~ \lim x_n\},$$ where $c$ is included in $\ell^\infty$. How can I find a function $T$ such that $\ker(T)=c$? Also, after that, how can I see if $T$ is continuous? Thanks.
1
vote
1answer
20 views

Operator norm on $M_n(A)$ where $A$ is a Banach algebra

On $M_n(\mathbb{C})$, if we take the operator norm by acting on $\mathbb{C}^n$ where $||(z_1,\ldots,z_n)||=\max_{1\leq i\leq n}||z_i||$, then for $[a_{ij}]\in M_n(\mathbb{C})$ we have ...
0
votes
0answers
29 views

Question about differentiability, definition and consequences.

Let $E,F$ be normed spaces, we say $f:E \to F$ is differentiable in $x_0\in E$ if there exist $Df(x_0) \in \mathcal{L}(E,F)$ such that $$\lim_{h\to 0}\frac{f(x+h)-f(x)-Df(x_0)(h)}{\|h\|}=0$$ or ...
1
vote
0answers
22 views

How can we check that for a given norm, we can found an inner product?

Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers. Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner ...
2
votes
2answers
19 views

Example of a weakly separable normed space that is not norm separable

I am looking for an example of a normed space which is separable with respect to the topology induced by all continuous linear functionals, but not separable with respect to the norm topology.
2
votes
1answer
33 views

Example for a norm on Hom(V,W) which is not determined by rank-one operators

Assume $(V,\|\cdot \|_V),(W\|\cdot \|_W)$ are two finite dimensional normed spaces (over $\mathbb{R}$). Any operator norm on $\text{Hom}(V,W)$ is determined by its value on rank-one operators. (This ...
3
votes
1answer
58 views

Is the supremum norm the only $ C^{*} $-norm on $ {C_{c}}(X) $, equipped with the usual pointwise operations?

Let $ X $ be a locally compact Hausdorff space. Then $ {C_{c}}(X) $ is a commutative $ * $-algebra with respect to addition, multiplication, scalar multiplication and conjugation (all pointwise ...
1
vote
1answer
40 views

existence of functionals

Let $X$ be a finite-dimensional normed space. Consider a non-empty convex set $C\subset X$ such that $0\notin C$. Notice that $C$ has a dense and countable subset $\{x_n\}$. $\forall n $ let $C_n= ...
4
votes
2answers
53 views

Can we reconstruct norms from their induced operator norm?

Assume $(V,\|\cdot \|_V),(W\|\cdot \|_W)$ are two finite dimensional normed spaces (over $\mathbb{R}$). Now I only give you the operator norm on $\text{Hom}(V,W)$ w.r.t $\|\cdot\|_V,\|\cdot \|_W$. ...
0
votes
0answers
15 views

If $S$ and $T$ are closed vector subspaces then $S+T$ is closed [duplicate]

Let $V$ be a Banach normed space, $S,T \subset V$ be closed vector subspaces. Assume $\operatorname{dim}(T)<\infty$. Show that $S+T$ is closed. So I encountered this problem trying to use ...
3
votes
1answer
21 views

Question regarding convergence in $L^p$ spaces.

When solving an exercise regarding $\ell^p$ spaces, I came up with the following question. The exercise said, Let $1<p\leq \infty$ and let $p'=\frac{p}{p-1}$. Let $b\in \ell^{p'}$ and define ...
0
votes
1answer
15 views

Bounded linear space (elementary question)

Does exist (nonzero) bounded normed space over any field? Fix normed linear space $L$ over field K. We have $x+x+x+\cdots\in L$ So $||nx||=n||x||\rightarrow \infty$ when $n \rightarrow \infty \\$ ...
1
vote
1answer
40 views

Find the norm of operator

Let we have the following operator $$T:\ell_p \rightarrow \ell_p$$ $T(ξ_1,ξ_2,ξ_3,ξ_4,ξ_5,ξ_6,ξ_7,ξ_8,ξ_9,ξ_{10},ξ_{11},ξ_{12},ξ_{13},ξ_{14},..)=(ξ_1,ξ_3,ξ_5,ξ_7,ξ_9,ξ_{11},ξ_{13},...)$ How can I ...
1
vote
1answer
33 views

When the group of isometries of a norm determines the norm?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $ Let $V$ be a finite-dimensional normed space. Assume that $G=\text{ISO}(||\cdot ||_1) = \text{ISO}(||\cdot ||_2)$. When can we conclude that ...
3
votes
1answer
17 views

orthogonality-like in vector normed space

Given a normed real vector space $V$, and a vector $x\ne 0$, there exists a vector $y\ne 0$ such that $\|x+y\|=\|x-y\|$? I know that it exists if the norm is induced by a scalar product (in ...
4
votes
1answer
46 views

A norm which is symmetric enough is induced by an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $ It is a fact that for every norm $\| \|$ on a finite dimensional (real) vector space, its isometry group $\text{ISO}(|| \cdot ||)$ is ...
0
votes
0answers
47 views

Calculate $\|T\|$

Let $k:[0,1] \times [0,1] \to \Bbb{R}$ continuous, and let $K:C[0,1] \to C[0,1]$ be defined as $$K(f)(x)=\int_{0}^{1}k(x,y)f(y)dy$$ Show that $K$ is a continuous linear operator, and bound ...
0
votes
1answer
24 views

Why can we consider elements of a normed space $X$ as elements of a normed space $Y$, if there is an embedding between these spaces?

Let $(X,\left|\;\cdot\;\right|)$ and $(Y,\left\|\;\cdot\;\right\|)$ be normed spaces and $\iota :X\hookrightarrow Y$ be an embedding. Often when I read that such an embedding $\iota$ exists, I read ...
0
votes
1answer
39 views

Question about norms $p$ and $q$.

I have a simply question: Show that if $x,y \in \mathbb{R^n}$, then $$\biggr|\sum{x_jy_j}\biggr|\leq \| x \|_p \| y \|_q$$ First, I proved that, if $s,t\geq 0$, then ...
0
votes
3answers
29 views

$T$ continuous in $x_0$ then $T$ is continuous

Let $T:V\to W$ be a linear operator, with $V, W$ normed spaces. Show that if there exist $x_0 \in V$ such that $T$ is continuous in $x_0$ then $T$ is continuous. I'm thinking that given an ...
0
votes
0answers
14 views

Invertible polynomials and polynomial norms

I am interested in normed rings, and I got to thinking about polynomial rings. In particular, if $R=k[x]$ is the ring of polynomials in one variable over a field $k$ (say characteristic 0), then the ...
1
vote
1answer
15 views

Existence of non continuous Linear transformation

Given that $M, N$ are normed linear space over same scaler, $\dim M=\infty, N\ne\{0\}$, we need to show existence of a linear transformation $T : M\to N$ which is not continuous. I am not having a ...
0
votes
1answer
7 views

$T$ induces a natural linear map $L : N/K\to M$

We are given that $T$ is a continuous linear map of a normed linear space $N\to M$, $M$ is also normed linear space. $K$ be it's Kernel, we need to show the title. I think here is the natural ...
0
votes
1answer
14 views

To find norm of $T, T(x) = x+M$

We are given that $M$ is a closed linear subspace of a normed linear space $N$, $T $ be the natural mapping from $N\to N/M$ as mentioned in title, we need to show that $T$ is continuous Linear ...
1
vote
0answers
49 views

Equivalence of two norms on $L^p(M)$, $M$ compact manifold.

Let $(M,g)$ be a compact Riemannian manifold, $\mu(g)$ the Riemannian Lebesgue measure. Then we can define the usual $L^p$-spaces (lets assume $p<\infty$), $L^p(M,g):=L^p(M,\mu(g))$. For $f\in ...
1
vote
2answers
55 views

How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$ [duplicate]

How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$ I need to use normed space axioms but I'm unable to figure it out. I think the scalar axiom and triangle inequality are helpful.
0
votes
2answers
29 views

Why is this function uniformly continuous?

Assuming $X$ is a normed space. Why is a function $f:K\rightarrow\mathbb{R}$ uniformly continuous on a subspace $K\subset X$, if $K$ is sequentially compact and $f$ is continuous?
2
votes
4answers
60 views

Show that if $\sum x_n$ converges then $x_n \to 0$

Let $(V,\|\|)$be a normed space. Let $(x_n) \subset V^{\Bbb{N}}$. We say that $\sum x_n$ converges if, $\lim_{n\to \infty} \sum_{i=1}^{n}x_i$ exists. Show that if $\sum x_n$ converges then ...
2
votes
1answer
39 views

$||f||_1 =(\int_a^b [|f|^2+|f'|^2]dx)^{1/2}$. Is this normed space complete?

Define $C_1^1[a,b]$ to be the space of continuously differentiable functions on $[a,b]$, with norm $$||f||_1 =\left(\int_a^b \left(|f|^2+|f'|^2\right) dx \right)^{1/2}$$ Is this normed space ...
1
vote
3answers
33 views

Do we need to have a subsequence such that $\lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a normed space and $(x_n)_{n\in\mathbb{N}}\subseteq X$. Can we prove that there is a subsequence ...
0
votes
0answers
34 views

How is $\|\cdot\|_1$ defined on a finite-dimensional real vector space?

Let $V$ a normed space over $\Bbb{R}$, and let $S$ be a finite dimensional subspace. I'm trying to show that $S$ is complete, I've already seen this question has ben made, but I have a precise doubt. ...
0
votes
3answers
61 views

Is a Banach space also a metric space?

Since a Banach space is a complete normed vector space and a norm always induces a metric, a Banach space must be a metric space, right? If so, why is a Banach space defined as a complete normed ...
0
votes
2answers
59 views

Is the unit sphere in an infinite dimensional Hilbert space closed?

Is a unit sphere in an infinite dimensional hilbert space closed. By the triangle inequality it is clear that the all the limit points of the sphere are inside the closed unit ball. But I cannot ...
5
votes
1answer
45 views

Is there a nowhere differentiable norm on $\Bbb R^n$?

Is there a nowhere differentiable norm on $\Bbb R^n$? The non differentiable norms that I know (e.g. the one norm $\|\cdot\|_1$ and the infinity norm $\|\cdot\|_{\infty}$) are non differentiable ...