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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1answer
773 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
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2answers
492 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
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0answers
8 views

For $A: Y \rightarrow Z$, find $z \in Z$ s.t. $z \notin A(Y)$

Let $Y =C[0,1]$ be the space of real-valued continuous functions equipped with the supremum norm $\|·\|_∞$, and $Z = \ell^∞$. $Af = (f(2^{−k}))_{k≥1}$ defines a bounded linear operator $A:Y →Z$. ...
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0answers
24 views

Normed space, inner product space, which is larger?

My textbook says normed space is inner product space unless it satisfies some parallelogram identity. In this case, we can conclude inner product space is a subset of normed space. However, norm is ...
4
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1answer
50 views

How do I link dimension of a normed vector space with closedness?

Let $X$ be a Normed Vector Space, for any $x\in X$ and $r>0$. Let $W:=\{y\in X : \|y-x\|\leq r\}$ and $S:=\{y\in X : \|y-x\|<r\}$ Prove: $W$ is closed if $\dim(X)<\infty$ I can't think of a ...
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0answers
39 views

A not so obvious corollary?

Let $X$ be a normed vector space over $\mathbb{K}$. Then there is only one completion of $X$, the banach space $\hat{X}$ such that $X$ is a dense subspace of $\hat{X}$. I am trying to prove that there ...
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1answer
50 views

Every normed space has a completion?

So I know that a completion of $X$ is a Banach space $Y$ such that $X$ is isometrically isomorphic to a dense subset of $Y$, say $A$. So we need to prove that we can always find a $T \in L(X,A)$ such ...
16
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1answer
5k views

Proof that every finite dimensional normed vector space is complete

Can you read my proof and tell me if it's correct? Thanks. Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm ...
0
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0answers
29 views

Show two norms are equivalent

Let $$N(z)=\Bigg(\sum_{n=1}^{\infty} \bigg|\frac{c_n}{n}\bigg|^3\Bigg)^{1/3}$$ be a norm on $\ell^3$ where $z=(c_n)_{n\geq1} \in \ell^3.$ Are the norms $N(\cdot)$ and $\|\cdot\|_3$ equivalent? ...
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0answers
10 views

Normed-Spaces and Integrals Question

Notations: $[f]$ is the equivalence class of $f$. $^\ast\int_{\mathbb{R}^n}f$ is the upper integral of $f$ $_\ast\int_{\mathbb{R}^n}f$ is the lower integral of $f$ Functionals ...
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2answers
21 views

How to show that a function in $L_2$ space is uniformly bounded in the “truncated norm” [on hold]

Let $$L_2[0, \infty) = \left\{f: \mathbb{R}_{+} \to \mathbb{R}^n \mid \int_0^\infty f(t)^Tf(t) \, dt< \infty \right\}$$ Define the truncated norm as $$\|f\|_K = \sqrt{\int_0^K f(t)^Tf(t) \, ...
0
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1answer
12 views

Expression related to dual norm on bounded linear functionals

Given a vector space $V$ and a norm $\|\cdot\|$ on $V$, the dual norm $\|\cdot\|^*$ on $V^*$ is given by $\|f\|^* = \sup \left\{\frac{f(v)}{\|v\|}\right\}$ over all nonzero vectors $v$. I've found ...
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0answers
24 views

Equivalence of two norms.

Suppose, that we have $||.||_1$ and $||.||_2$ norms defined on an arbitrary $X$ vector space. $X$ is a complete space with both of them. For all $x_n, n \in \mathbb{N} \subset X$ series, if $\lim_{n ...
5
votes
2answers
48 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
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1answer
80 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
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0answers
19 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
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0answers
12 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 ...
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3answers
46 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 \| = ...
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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
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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
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1answer
473 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
107 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
941 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. ...
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0answers
25 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
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2answers
26 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
29 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 ...
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0answers
26 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 , ...
3
votes
1answer
32 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
140 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
31 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
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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
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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 ...
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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.
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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
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0answers
21 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
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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 ...
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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
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2answers
374 views
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1answer
16 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 ?
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1answer
26 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
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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
11 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
11 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 $Y$ ?