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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21 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 ...
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2answers
16 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
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
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1answer
57 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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1answer
39 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= ...
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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 $\|\cdot\|_V,\|\cdot \|_W$. ...
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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 ...
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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
31 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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1answer
32 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
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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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1answer
45 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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47 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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2answers
52 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
57 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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3answers
32 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 ...
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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. ...
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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 ...
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2answers
54 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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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 ...
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2answers
87 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 ...
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1answer
20 views

Book reference for introduction to Normed Spaces

I wanted a book reference for the study of normed spaces and linear operators. Im still not into functional analysis, but I wanted a reference of an introductory book as to start reading. What would ...
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2answers
60 views

Triangle Inequality for norm integral $\|f\|_1=(\int_a^b [|f|^2+|f'|^2]\mathsf dx)^{1/2}$.

Define $C_1^1[a,b]$ to be the space of continuously differentiable functions on $[a,b]$, with norm $$\|f\|_1=\left(\int_a^b[|f|^2+|f'|^2]\mathsf dx\right)^{1/2}.$$ Show that this is a proper ...
1
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2answers
48 views

Is $\ell_{\infty}$ a smooth normed space?

A normed linear space $X$ is said to be smooth if for each $x \in X$ there exists a unique functional $x^* \in X^*$ with $\|x^*\|=1$ such that $x^*(x)=\|x\|$. I know that $L_1[0,1]$ is not smooth. ...
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1answer
24 views

Compute the limit of this expression of norms:

Compute the limit, as n goes to infinity, of the quotient: $$\frac{||A^{n+2}(x)||}{||A^n(x)||} $$, given the matrix $$ \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ ...
3
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0answers
36 views

Calculating Norms for the Transpose

How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$? Notation. ...
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1answer
24 views

Find two functions in $L_p(\Bbb R)$, whose product $f\cdot g$ does not belong to $L_p(\Bbb R)$.

How can I find two functions in $L_p(\Bbb R)$, with their product $f\cdot g$ not belonging to $L_p(\Bbb R)$?
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1answer
18 views

Differentiability in normed spaces

I really need a help with the following exercise: Suppose $\mathbb{E}$ and $\mathbb{F}$ are normed spaces, $A \subseteq \mathbb{E}$ is an open set, $f: A \to \mathbb{F}$ is differentiable on $A$, and ...
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1answer
23 views

Taking limits of norms of a matrix raised to the nth power:

Given a matrix $$ A = \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ \end{bmatrix} $$ and a vector $x = \begin{bmatrix}1&0\end{bmatrix}$, compute ...
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1answer
19 views

Prob. 11, Sec. 2.10 in Erwin Kreyszig's book

If $X$ is a normed space and dim $x=\infty$, show that the dual space $X'$ (set of all bounded linear functionals on $X$) is not identical with algebraic dual space $X^*$ (set of all linear ...
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25 views

Prob. 10, Sec. 2.10 in Erwin Kreyszig's book

Here is my question: Let $X$ and $Y\neq\{0\}$ be normed spaces, Where dim $X = \infty$. Show that there is at least one unbounded linear operator $T:X \mapsto Y$. (use a Hamel basis). This ...
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4answers
57 views

does $\|A\| > \|B\|$, $A$ and $B$ matrices, imply that $\|Ax\| >\|Bx\|$ for all $x$ in some vector space?

I'm wondering if I can prove the first inequality in a question that I am working on, does that make the second inequality automatic? Thanks,
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1answer
35 views

Convergence of sequence of function in norm.

Let $1\leq p<\infty$. Suppose that $\{f_k\}$ is a sequence in $L^p(X,\mathcal{M},\mu)$ such that the limit $f(x)=\lim_{k \to \infty}f_k(x)$ exists for $\mu$-a.e. $x\in X$. Asumme that ...
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2answers
44 views

Prob. 6, Sec. 2.10 in Erwin Kreyszig's functional analysis.

Definition (Dual space $X'$). Let $X$ be a normed space. Then the set of all bounded linear functionals on $X$ constitutes a normed space with norm defined by $$ \left \| f \right \|= ...
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0answers
37 views

Prob. 4, Sec. 2.10 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Now here's Prob. 4, Sec. 2.10 in Introductory Functional Analysis With Applications by Erwin Kreyszig: Let $X$ and $Y$ be normed spaces and $T_n:X\rightarrow Y\ (n=1, 2, \cdots)$ bounded linear ...
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1answer
24 views

Prob. 10, Sec. 4.2 in Kreyszig's functional analysis book: There is a linear functional for every sublinear functional …

If $p$ is a sublinear functional on a real vector space $X$, then there exists a linear functional $\tilde{f}$ on $X$ such that $-p(-x) \leq \tilde{f}(x) \leq p(x)$ for all $x \in X$. How to prove ...
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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 ...
0
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1answer
12 views

Is $\inf_{y\in Y} \|\alpha x-y\|=\inf_{y' \in Y} \|\alpha x-\alpha y'\|$

In my functional analysis homework problem I was trying to a extend a linear functional from a proper sub-space $Y$ to the whole space $X \supset Y$using the Hahn Banach theorem. In order to do that I ...
0
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2answers
35 views

Prob. 6, Sec. 4.2 in Kreyszig's functional analysis book: Continuity of a subadditive functional at zero implies continuity

Let $X$ be a real normed space, and let $p \colon X \to \mathbb{R}$ be a functional such that $$ p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$ and such that $$p(\theta) = 0, \ \mbox{ ...