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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5
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2answers
41 views

For a normal operator is it true that $\|T^*T^2\| = \|T^3\|$?

For a normal operator is it always true that $\|T^*T^2\| = \|T^3\|$? See the accepted answer for the case in a Hilbert space Update: how about $\|T^*T^2\| = \|T\|^3$ in a Hilbert space
5
votes
1answer
78 views

$f_1,…,f_n$ be linear functionals on a real vector space $V$, then is there a norm on $V$ which makes every $f_i$ continuous?

Let $V$ be a real vector space, $f_1,...,f_n$ be linear functionals on $V$; then does there exist a norm on $V$ with respect to which each of $f_i$ is continuous? And what if we have infinitely many, ...
4
votes
0answers
15 views

Can we detect smoothness of a norm by its behavior along paths?

We say a norm $\| \cdot \|$ on $\mathbb{R}^n$ is smooth if it is smooth as a function $\mathbb{R}^n\setminus \{0\} \to \mathbb{R}$. (i.e, after restricting the domain). We say a norm is smooth along ...
0
votes
0answers
9 views

Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda ...
1
vote
3answers
44 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x ...
0
votes
1answer
24 views

$| \langle a , i \rangle| \leq \| a\|$ if $\|i\|=1$ this space is a normed vector space upon $\langle , \rangle$ . Why does this apply?

I tried over Cauchy Schwarz to conclude, but could not. Anyone see why this is ? The term: normed vector space upon $\langle , \rangle$ i hear for the first time, Im assuming it means that: $$\|a \| = ...
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
24 views

When is injective contraction isometry

Let $f:X \to Y$ be a injective linear map between (semi-)normed spaces, s.t $B_Y = f(B_X)$, $B_X,B_Y$ being the unit balls. Is $f$ an isometry? If so, was there a superfluous requirement? EDIT: I ...
1
vote
1answer
467 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
2
votes
2answers
148 views

How does one prove $\left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert$?

Can the triangle inequality for norms works for infinite some as well? Hope that someone can give some hints to prove it or give a counterexample. Thanks!
0
votes
1answer
104 views

Goldstine theorem and weak topology on $E$ induced by weak* topology on $E''$

In the book where I'm studying there this exercise: "(Goldstine's theorem). Recall that $E \subset E''$, or, more precisely, $J_E(E)$ is a subspace of $E''$ where $J_E :E \longrightarrow E''$ is ...
4
votes
5answers
924 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
0
votes
1answer
18 views

Equivalent particular norms, can you point the right direction?

Let $P([0,1])$ be the space of all the polinomial with complex entries, defined in $[0,1]$ . Show that $||f||_\infty=sup_{t\in [0,1]}|f(t)|$ and $||f||_1= \int_{0}^{1}|f(t)|dt$ are equivalent norms. ...
0
votes
0answers
24 views

Spectrum of Linear Operator

The spectrum of a linear operator on a finite dimensional space is pure point spectrum, that is, both continuous and residual spectrums are empty. Or we can say that on a finite dimensional space, ...
0
votes
2answers
25 views

Let $C$ be a convex closed nonempty subset of a Hilbert space $H$. Show that there is a unique element in $C$ with the minimum norm.

Let $H$ be a Hilbert space and $C \subset H $ be a convex, closed and nonempty subset of $H$. Prove that there exists a unique $x_0\in C$ with minimum norm among the elements of $C$. I don't know ...
2
votes
2answers
28 views

Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
0
votes
0answers
25 views

Proving that these functionals are bounded, and finding their norms.

Proving that these functionals are bounded, and finding their norms. $$a.)f_1(x):c_o \to \mathbb R , f_1(x)=\sum_{n=1}^{\infty}\frac{x_n}{2^{n-1}} \\ b.)f_2(x):l_1 \to \mathbb R , ...
2
votes
1answer
30 views

Showing a certain map is a norm.

Define $$\|x\|=\sqrt[3]{(|x_1|^2+|x_2|^2)^{3/2}+|x_3|^3}$$ for any $x=(x_1,x_2,x_3)\in \Bbb R^3$. Show that $\|\cdot\|$ is a norm on $\Bbb R^3$. I stuck on showing triangle inequality. I don't know ...
2
votes
1answer
40 views

$|v|_{2,\Omega}=0$ implies $v=0$

I am stuck on this computation: let $\Omega$ be a domain in $\mathbb R^2$ and let $\Gamma_0$ be a relatively open proper subset of $\Gamma:=\partial\Omega$. Define $$ V=\{v \in H^2(\Omega); ...
2
votes
3answers
134 views

Weak convergence $\iff$ strong convergence in finite dimensional space

I am seeking a proof of the following claim. Weak convergence $\implies$ strong convergence in a finite-dimensional normed linear space. Thank you.
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0answers
28 views

Finding $\|f\|$

Given $f: \ell^2 \rightarrow \mathbb R$ defined by $$f(a_1,a_2,...)=\sum_{n=1}^{\infty} \frac{(-1)^n}{3^{n-1}} a_n$$ Is $f$ a bounded linear functional? If yes, then find $\|f\|$. It is definitely ...
1
vote
1answer
39 views

Symmetric norms on $\mathbb{R}^n$ are between the $\ell^1$ and $\ell^\infty$ norms

We say that a norm $\|\cdot\|$ on $\mathbb{R}^n$ is symmetric if it is invariant under sign changes and permutations of the components. Suppose that $$\|(1, 0, ..., 0)\| = 1$$ for a symmetric norm. ...
0
votes
1answer
27 views

Clarification about completeness of metric spaces

This is probably a very silly question but it bothers me for some time. We define a metric space $X$ to be complete if every Cauchy sequence in $X$ converges to some point in $X$. But any metric ...
7
votes
5answers
1k views

Fixed point Exercise on a compact set

Let $K$ a compact normed space and $f:K\rightarrow K$ so that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ Prove that $f$ have a fixed point.
1
vote
2answers
35 views

On the dimension of a real Normed Linear Space possessing a certain property

Let $X$ be a real NLS such that for every proper subspace $Y$ of $X$ , $\exists x \in X$ such that $||x||=1$ and $dist (x,Y)=1$ ; then is $X$ finite dimensional ?
2
votes
0answers
20 views

Find value of $p$ such that $\sum |b_n|^p $ converges

Let $(X , \|\cdot \|)=(\ell^2, \|\cdot \|_2)$. Let $e_n \in \ell^2$. Fora bounded linear functional $\phi$ on $X$, let $b_n=\phi (e_n)$ for each $n \geq 1$. Find the value of $p \geq 1$ such that the ...
2
votes
2answers
43 views

Linear functional is continuous $\implies$ it is bounded

Let $f:X \rightarrow \mathbb R$ be a continuous linear functional. Prove $f$ is bounded. Since it is continuous, $\forall \varepsilon >0$, there exists $\delta >)$ such that ...
0
votes
0answers
19 views

Showing a subspace is closed

Let $X = (\textbf{c}_{0},\|\cdot\|_{\infty})$ and $Y$ be defined as $$ Y := \bigg\{ \{x_{i}\} \in \textbf{c}_{0} : \sum_{i=1}^{\infty} \frac{x_{i}}{2^{i}} = 0 \bigg\}. $$ (1) Show that ...
3
votes
2answers
370 views
0
votes
1answer
15 views

$X,Y$ be real NLS ; $T:X \to Y$ be a linear map such that $\ker T$ is closed ; then does $T$ have closed graph?

Let $X,Y$ be real normed linear spaces and $T:X \to Y$ be a linear map with closed kernel ; then does $T$ have closed graph ? What if we assume arleast one of $X,Y$ to be complete ?
0
votes
1answer
25 views

Show that an open linear mapping between normed spaces is surjective

I'd just like to know where to begin. The exact thing to prove: Let $X$ and $Y$ be normed spaces and $R:X\to Y$ is an open linear mapping. Show that $R$ is surjective. And to be clear, neither of the ...
0
votes
2answers
21 views

$X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ? ...
1
vote
1answer
10 views

$Y$ is a ( closed) proper subspace of a real NLS $X$ such that $dist (x,Y)=1$ for some $x \in X$ with $||x||=1$ ; is $Y$ finite dimensional?

Let $Y$ be a finite dimensional proper subspace of a real NLS $X$ , we know that we can find $x\in X$ ( depending on $Y$) , such that $||x||=1$ and $dist (x,Y):=\{||x-y||:y\in Y\}=1$ . I would like to ...
2
votes
0answers
20 views

$f \in \mathcal l^{\infty}{'} $ ; $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence of non-negative terms ; is $f$ bounded? [duplicate]

Let $f:\mathcal l^{\infty} \to \mathbb R$ be a linear functional such that $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence with non-negative terms ; then is $f$ continuous ?
0
votes
1answer
12 views

Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ?

Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ? If $y=x$ then it is easy to see that that's the case . So I thought let $y \ne x$ ; I tried ...
0
votes
0answers
10 views

Extended continuous linear transformation keeping same norm , when co domain is finite dimensional

Let $Y$ be a subspace of a real normed linear space $X$ , $T:Y \to \mathbb R^n$ be a continuous linear transformation ; then can we extend $T$ to a continuous linear transformation $\bar T : X \to ...
0
votes
1answer
15 views

$X$ be Banach space with a proper dense subspace $Y$. Can the identity operator on $Y$ be extended to a continuous function from $X$ into $Y$?

Let $X$ be any Banach space with a proper dense subspace $Y$. Can the identity operator on $Y$ be extended to a continuous function from $X$ into $Y$ ?
1
vote
1answer
20 views

Is it possible to extended finite rank continuous linear transformation to a continuous linear transformation with same range?

Let $X,Y$ be normed linear spaces , $W$ be a linear subspace of $X$ , let $T:W \to Y$ be a continuous linear tranformation with finite rank i.e. $T(W)$ is finite dimensional ; then can we extend $T$ ...
0
votes
1answer
20 views

Show $c_{00}$ is not closed under supremum norm

Show $Y=c_{00}$ is not closed under $(\ell^{\infty}, \|\cdot\|_{\infty})$. I know that I need to find a $(y_n) \in c_{00}$ such that this converges to $y$ with $y \notin c_{00}$. So we need $\|y_n ...
0
votes
0answers
13 views

Finding norms on a piecewise function

For each $n=1,2,...$ let the function $g_n \in C[0,1]$ be defined by \begin{equation} g_n(t)=\begin{cases} 2nt & 0 \leq t \leq 1/2n \\ 2-2nt & 1/2n \leq t \leq 1/n \\ 0 & 1/n \leq t \leq ...
1
vote
0answers
19 views

Show $Af = (f(2^{−k}))_{k≥1}$ is bounded and find $\|A\|$

Let $Y =C[0,1]$ be the space of real-valued continuous functions equipped with the supremum norm $\|·\|_∞$, and $Z = \ell^∞$ be the space of bounded sequences of real numbers equipped with its usual ...
5
votes
0answers
42 views

Proving linear operator is bounded

Prove that the formula $T(b_1,b_2,b_3,...,b_n,...) = (b_1, b_2/2 ,..., b_n/n ,...)$ defines a bounded linear operator $T : (ℓ^∞,∥·∥_∞)→(ℓ^∞,∥·∥_∞)$. Proving that it is linear is easy. Need help with ...
9
votes
1answer
2k views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
0
votes
1answer
48 views

Isometry and isomorphism normed spaces

Problem. Let $X$, $Y$ be real normed vector spaces and $ f $ isometry space $ X $ in the space $ Y $. Show that there is isomorphism $ A $ spaces $ X $ on the space $ Y $ and vector $ c \in Y $ such ...
0
votes
1answer
25 views

$X,Y$ be Banach , $T \in \mathcal B(X,Y)$ be onto ; then , for every sequence $y_n \to y \in Y$ , $\exists x_n \to x\in X$ s.t. $T(x_n)=y_n , T(x)=y$?

Let $X,Y$ be Banach spaces , $T:X \to Y$ be a surjective continuous linear transformation , then is it true that for every convergent sequence $\{y_n\}$ in $Y$ , converging to $y \in Y$ , there exist ...
0
votes
1answer
31 views

What are the differences between $l^p$ space and $L^p$ space?

I always thought small $l^p$ space, or so-called Banach space, is the space equipped with a norm similar to vector norm as $$||x||_p = \left( \sum_{i\in\mathbb{N}} |x_i|^p \right)^{\frac{1}{p}}$$ ...
0
votes
1answer
26 views

The image of linear operator, $T(\ell ^{\infty})$

$T:(\ell^\infty, \|\cdot\|_\infty) \rightarrow (\ell^\infty, \|\cdot\|_\infty)$ with $T(b_1,b_2,\ldots)=(b_1, b_2/2, b_3/3,\ldots)$ is a bounded linear operator. Show that $w = (1, 1/\sqrt2, 1 ...
0
votes
0answers
39 views

$Y,Z$ be linear subspaces of a Banach space $X$ ; $Y$ be finite dimensional , $Z$ closed in $X$ ; is $Y+Z$ closed in $X$? [duplicate]

Let $Y$ and $Z$ be linear subspaces of a Banach space $X$ , such that $ Y$ is finite-dimensional and $Z$ is closed in $X$ , then is $Y +Z$ also closed in $X$ ?
1
vote
0answers
23 views

The concept of Subspace of a Normed Vector Space

I am working with Banach Spaces, which are complete Normed Vector Spaces (NVS). The norm on a NVS $(E, ||\cdot||_1)$ defines a metric which in turn defines a topology. Now let us consider $F ...
3
votes
1answer
51 views

Characterizing orthogonally-invariant norms on the space of matrices

Denote by $M_n$ the space of $n \times n$ real matrices. We say a norm on $M_n$ is orthogonal invariant if: $$\|OX \|=\| XO\|=\|X \| \, \, \forall O \in O_n,X \in M_n$$ I am trying to characterize ...