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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45 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F ...
0
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1answer
52 views

Prob. 8, Sec. 4.5 in Kreyszig's functional analysis book: The inverse of the adjoint operator is the adjoint of the inverse operator

Let $X$ and $Y$ be normed spaces, both real or both complex, let $B(X,Y)$ denote the space of all the bounded linear operators $T \colon X \to Y$, and let $T^\times$ denote the adjoint operator of ...
2
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1answer
87 views

How to apply Theorem 4.3-3 in the proof of Theorem 4.5-2 in Kreyszig's functional analysis book?

Here's Theorem 4.3-3 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space and let $x_0 \neq 0$ be any element of $X$. Then there exists a bounded ...
2
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2answers
67 views

Compact linear operators $S$ and $T$, show that $S(I-T)=I$ if and only if $(I-T)S=I$ and deduce that $I-(I-T)^{-1}$ is a compact operator

If $T:X\to X$ is a compact linear operator, then for any bounded linear operator $S:X\to X$ we have that $S(I-T)=I$ if and only if $(I-T)S=I$. Where $X$ is a normed space, also $T$ is bounded. With ...
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1answer
43 views

Manipulating sets ($+$, etc).

I was seeing a proof of the Open Mapping Theorem, in Kreyszig's book, and I have no problems with it. But there's a point in which he does something like: $$\begin{align}B_Y(0,r) \subset ...
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1answer
35 views

$p$-operator space property

If $S,T,U,V\in B(L_p(X,\mu))$, $p\in[1,\infty)$, and we regard $\begin{pmatrix} S & T \\ U & V \end{pmatrix}$ as an operator on $B(L_p\oplus_p L_p)$, then supposedly we have ...
0
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1answer
23 views

Comparing norms - $M_{kn}(A)$ versus $M_k(M_n(A))$

I am having to deal with the problem of passing between $M_{kn}(A)$ and $M_k(M_n(A))$ where $A$ is a Banach algebra, removing parentheses in one direction, and adding parentheses in the opposite ...
0
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2answers
49 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 = ...
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1answer
44 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 ...
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1answer
32 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 ...
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1answer
45 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
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1answer
24 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 ...
3
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0answers
43 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 ...
0
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0answers
29 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
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1answer
54 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?
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1answer
36 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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2answers
51 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
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2answers
63 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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1answer
50 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
34 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
108 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
34 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
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0answers
31 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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0answers
39 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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2answers
26 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
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1answer
40 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
74 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 ...
6
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0answers
54 views
+100

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, ...
1
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1answer
41 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
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2answers
61 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
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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
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1answer
29 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
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1answer
19 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
38 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
20 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
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1answer
52 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
48 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
28 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 ...
0
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3answers
39 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
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0answers
18 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 ...
0
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1answer
22 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
12 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
20 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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0answers
63 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
128 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
41 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
67 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
2answers
56 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
51 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 ...