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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7 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 ...
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1answer
25 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 $Hom(V,W)\neq L(V,W)$.
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1answer
32 views

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

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?
3
votes
2answers
36 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 ...
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2answers
43 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 $0$, ...
-2
votes
1answer
40 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.
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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"
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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.
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1answer
49 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 ...
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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 ...
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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 ...
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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
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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 ...
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votes
2answers
17 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.
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1answer
25 views

Prob. 8, Sec. 4.2 in Kreyszig's functional analysis book: Nonnegativity of a subadditive functional outside a sphere implies nonnegativity

If a subadditive functional $p$ defined on a normed space $X$ is non-negative outside a sphere $\{ \ x \in X \ \colon \ \Vert x \Vert = r \ \}$, then how to show that $p$ is non-negative for all $x ...
3
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1answer
91 views

How to show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete

Show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete. I am not sure where to even start with this question. I'm quite sure it involves Cauchy and also Bolzano-Weierstrass but I'm not sure ...
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 ...
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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 $\|\cdot\|_V,\|\cdot \|_W$. ...
1
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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= ...
0
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0answers
14 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
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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 ...
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1answer
14 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 $n \rightarrow \infty \\$ ...
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1answer
32 views

Find the norm of operator

Let we have the following operator $$T:L^p \rightarrow L^p$$ $T(ξ_1,ξ_2,ξ_3,ξ_4,ξ_5,ξ_6,ξ_7,ξ_8,ξ_9,ξ_{10},ξ_{11},ξ_{12},ξ_{13},ξ_{14},..)=(ξ_1,ξ_3,ξ_5,ξ_7,ξ_9,ξ_{11},ξ_{13},...)$ How can I find the ...
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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 ...
2
votes
1answer
163 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
1
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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 ...
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 ...
3
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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 ...
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0answers
48 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 ...
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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
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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
28 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 ...
5
votes
4answers
261 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
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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
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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 ...
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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 ...
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1answer
58 views

In a normed vector space $(V,\lvert . \rvert)$ show that $f:V\rightarrow \mathbb{R}$ with $f(v)=\lvert v\rvert$ is uniformly continuous

In a normed vector space $(V,\lvert . \rvert)$ show that $f:V\rightarrow \mathbb{R}$ with $f(v)=\lvert v\rvert$ is uniformly continuous The first part of the question says to prove the "reverse ...
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vote
1answer
44 views

Showing that a map $x \to \|x\|$ is continuous?

I am given this: Consider a real Banach space $X$ with norm $\|*\|$. 1) Show that the map $x\to \|x\|$ from $X$ to $\mathbb{R}$ is continuous. Is it uniformly continuous? 2) Show that the maps ...
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2answers
53 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.
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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?
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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
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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 ...
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votes
2answers
56 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 ...
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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 ...
9
votes
2answers
88 views

Isometry group of a norm is always contained in some Isometry group of an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $Let $||\cdot||$ be a norm on a finite dimensional real vector space $V$. Does there always exist some inner product $\<,\>$ on $V$ such ...
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 ...
2
votes
1answer
109 views

Are these two definitions for dual norm equivalent

Suppose there is a $p$-norm $\left\|\cdot\right\|_p$, the dual function at $z$ of it is, $$\sup\left\{ z^{T} x-\left\|x\right\|_p:x\in\mathbb{R}^n\right\}$$ The second is, $$ \sup\left\{ z^{T} x : ...
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 ...