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
29 views

plane angle calculation problem

in calculating the angle between the plane $2x + y -2z +4 = 0$ and $z$ axis I got that the angle between the normal and $z$ axis is $131.81$. however if I take $90°$ minus that I get a negative angle ...
-2
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2answers
46 views

How can I prove this operator is not continuos

Let $X$ be the normed space of all polynomials on $[0,1]$ such that $\| x \| = \max \limits _{t \in [0,1]} |x(t)|$ and we have the following operator $Tx(t)=x'(t)$. Prove this operator is not ...
2
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1answer
81 views

Proving a set is open.

Let $E,||\cdot||$ be a finite dimensional normed space over $\mathbb R$. Let $U$ be an open subset of $E$ and $a\in U$ Let $A=\{x\in E \;|\; \forall t\in [0,1], (1-t)a+tx\in U\}$ ...
5
votes
1answer
135 views

When does continuous (or uniformly continuous ) function between normed linear spaces carries bounded sets to bounded sets ?

I know that if $f:\mathbb R^m \to \mathbb R^n$ is continuous then $f$ carries bounded sets to bounded sets . What if we say $X,Y$ are normed linear spaces and $f:X \to Y$ where $f$ is continuous ? ...
3
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0answers
46 views

Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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0answers
20 views

Mahalanobis distance to the ellipsoid center

I am confused about the following description: If we parameterize the ellipsoid $E$ as: $E = \{x|\ ||Ax-b||_2 \leq 1\}$. $A \in S_{++}^n$ Then the Mahalanobis distance to the ellipsoid center is ...
4
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2answers
63 views

Convergence Problem in Normed Space

Probably easy, but I'm stuck atm: A sequence converges in norm 1 if and only if it converges in norm 2, for all sequences. Are the two norms necessarily equivalent?
6
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0answers
88 views

One more AC equivalence question

Is "Every vector space admits a norm" weaker than AC? I know that the statement follows from "Every vector space has a basis", which is equivalent to AC.
0
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0answers
22 views

Convergence of a sequence in a normed vector space [duplicate]

help with homework problem... I feel like its easy, I'm just missing something Show that $\{||x_k||\}$ converges in $\mathbb{R}$ if $\{x_k\}$ converges in a normed vector space V. merci :) its ...
4
votes
3answers
198 views

If $\|u\|^2 = c^2$, then $\|\dot x\|^2 \to a $ for $\ddot x = -\dot x + u(t)$?

I am dealing with the next simple equation $$ \ddot x = -\dot x + u(t), $$ where $u, x\in\mathbb{R}^m$, with $m \geq 1$, and I am wondering if for $\|u\|^2 = c^2 > 0$ then $\|\dot x\|^2\to a$, ...
0
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1answer
68 views

Completion of a vector space inside a given Banach space

Let $X$ be a normed vector space and $Y$ be a Banach space such that $X$ is continuously embedded into $Y$ (this will be denoted by $X\hookrightarrow Y$ in the sequel). Is it always possible to find ...
3
votes
1answer
54 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
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1answer
26 views

Dimension of quotient normed linear space

Suppose $M$ is a normed linear space. $L$ and $N$ are two closed subspaces of $M$ such that $L \subseteq N$. Then $L$ is a closed subspace of $N$. Let $\text{dim}(M/N)=r$ and $\text{dim}(N/L)=s$. My ...
2
votes
1answer
53 views

Show that $X$ is Banach space and describe $X^*$.

Let $X=L^2(\mu)\times L^2(\mu)=\{(f,g)|f,g\in L^2(\mu)\}$ be the linear space normed by $\|(f,g)\|=(\|f\|_2^3+\|g\|_2^3)^{1/3}$. Show that $X$ is Banach space and describe $X^*$. My Work: We ...
2
votes
1answer
40 views

Show that given $ϵ>0,$ there exists $x∈X$ such that $∥x∥=1$ and $d(x,M)>1−ϵ.$

Let $X$ be a normed linear space and $M$ be a proper closed linear subspace of $X$. Show that given $ϵ>0,$ there exists $x∈X$ such that $∥x∥=1$ and $d(x,M)>1−ϵ.$ My Work: Let $ ϵ>0$. Since ...
2
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0answers
38 views

Exercise in Hahn-Banach Theorem; Finding linear functional $-p(-x)\leq f(x)\leq p(x)$

(The following exercises are in Kreyszig's book 218 page; EXE 10) I want to solve the following exercise : If $X=l^\infty$, let $p(x)=\lim\sup x_i $, whichi is sublinear. Then find a linear functional ...
2
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1answer
92 views

About a technique used in the proof of Hahn-Banach Theorem

Recall Hahn-Banach (cf. Kreyszig's book) : If $X$ is a real vector space with a sublinear functional $p$ and if $f$ is linear on a subspace $Z$ with $p(z)\geq f(z),\ z\in Z$, then there exists an ...
2
votes
2answers
46 views

prove that $\{Lx:\|x\|\leq 1 \}=\mathbb{C}$

Let $X$ be a linear normed space over $\mathbb{C}$. If a linear functional $L$ on $X$ is not continuous, prove that $\{Lx:\|x\|\leq 1 \}=\mathbb{C}$ Clearly $\{Lx:\|x\|\leq 1 \}\subseteq ...
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1answer
36 views

How does finite linear combinations of the $x_n$'s looks like?

Let $X$ be a normed linear space and let $\{x_n\}\subseteq X$. Prove that $x\in X$ is the limit of finite linear combinations of the $x_n$'s iff $Lx=0$ for all continuous linear functionals $L$ on ...
2
votes
0answers
81 views

Question about Stone-Weierstrass theorem

I have a question about Stone - Weierstrass theorem. In the space $C[0,2\pi]$ of continuous functions on $[0,2\pi]$ with the sup norm. Consider the spaces $M$ of all trigonometric polynomials. It's ...
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0answers
21 views

A question involving normed spaces and strictly convex spaces

Let $(X, \| \cdot \|_X)$ be a normed space and let $\| \cdot \|$ be a norm on $X$ such that $(X, \| \cdot \|)$ is strictly convex. How can I find a strictly convex space $(Y, \| \cdot \|_Y)$ and a ...
2
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0answers
54 views

Equivalence of norms between two Banach spaces

I just learnt open mapping theorem. And I met a statement online asserting that If $X$,$Y$ are Banach space, and $T:X\to Y$ be a continuous bijection, then norms for $X,Y$ are equivalent. Can we ...
1
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1answer
28 views

A question involving norms

Let $(X, \| \cdot \|_X), (Y, \| \cdot \|_Y)$ be normed spaces and $T : X \rightarrow Y$ a bounded operator. Let $x, y \in X$ and let the norm on $X$ $$ \|x\| = \|x\|_X + \|Tx\|_Y. $$ I can't show that ...
0
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1answer
22 views

$f \in \mathcal{C}_{0}(X) \Rightarrow \sup_{x \in X} |f(x)| = \max_{x \in X} |f(x)|$

For a topological space $X$ we define $$\mathcal{C}_{0}(X) : = \left\lbrace f \colon X \longrightarrow \mathbb{C} \ \text{continuous} \colon \forall \, \varepsilon >0 \ \exists \, K \subseteq X \ ...
1
vote
1answer
49 views

Why if $T$ is not a bounded operator then exists $ (x_n) $ that converges to $ 0_{X} $ for which $ \| T(x_n) \| \geq n^2 $ for all $ n $?

Let $X$ and $Y$ be normed spaces. Suppose that $ T: X \to Y $ is a linear operator and assume that $T$ is not bounded. Why with these assumptions can I say that exists a sequence $ (x_{n})_{n \in ...
1
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1answer
52 views

$\|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}}$

I want to prove that $$ \|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}} $$ I proved it by Holder inequality. But this is an exercise under "Interpolation". So I guess it can be proved using ...
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0answers
25 views

Meaning of Normed Space

I have the following notations: 1.$(L^1\bigcap L^\infty)(0,A)$. 2.$L^\infty ((0,A)^2)$ 3.$L^\infty(Q)$ where Q =$(0,T) \times (0,A)$. Can someone explain to me what does the L norm represents, their ...
1
vote
2answers
35 views

For $X \in M(n,\mathbb R)$ , let $||X||:=\sqrt{Trace(AA^t)}$ , then $||AB|| \le ||A||\space||B|| , \forall A,B \in M(n,\mathbb R)$?

Let $M(n,\mathbb R)$ be the set of all square matrices of size $n$ with real entries . For $A \in M(n,\mathbb R)$ , let $||A||:=\sqrt{Trace(AA^t)}$ , then is it true that $A,B \in M(n.\mathbb R) ...
1
vote
1answer
72 views

A Cauchy sequence has a rapidly Cauchy subsequence

I am trying to fill in the details of a proof related to the Riesz-Fischer Theorem. We need to show that every Cauchy sequence $\{f_n\}$ has a rapidly Cauchy subsequence. My text claims that we can ...
0
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0answers
50 views

Proving finite dimensional normed linear space is complete , without using equivalence of norms on finite dimensional vector spaces

Every finite dimensional normed linear space , over the field of real numbers , is complete . I know a proof of this result by using "every norm on a finite dimensional real vector space is equivalent ...
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2answers
54 views

Is the parallelogram equality satisfied in $l^1$?

I can't show that the parallelogram equality is satisfied / or is not satisfied in $l^1$. If $(v_n), (w_n) \in l^1$, then we have $$ || (v_n) + (w_n) ||_1^2 + || (v_n) - (w_n) ||_1^2 = || (v_n + ...
2
votes
1answer
44 views

Determining whats the Induced Metric

I have the normed space $({\rm Lip}([0,1]), \|\cdot\|)$, where ${\rm Lip}([0,1])$ is all Lipschitz functions from $[0,1]$ to $\Bbb R$, and $$\|f\|=|f(0)|+\sup_{0\le x,y\le ...
12
votes
1answer
191 views

In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

$(V,\|\cdot|)$ be a real normed linear space such that $\|x\|=\|y\|$ implies $\lim\limits_{n\to\infty} \|x+ny\|-\|nx+y\|=0$, then is it true that the norm comes from an inner-product space ?
1
vote
1answer
38 views

A real vector space is an inner product space if every two dimensional subspace is an inner product space ?

Is it true that a vector space over the field of real numbers is an inner product space if every two dimensional subspace is an inner product space ? does it have anything to do with Neuman-Jordan's ...
0
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0answers
30 views

Looking for a simpler proof of Day's characterization of inner-product spaces and related things

I know the theorem that if $(V,||.||)$ is a real normed linear space such that the parallelogram identity $||x+y||^2+||x-y||^2=2(||x||^2+|y||^2)$ holds , then the norm comes from an inner-product ...
0
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1answer
47 views

Showing a Set is Dense in $C[a,b]$

Show that $C_0[a,b]:= \{f \in C[a,b] : f(a)=f(b)=0 \}$ is $\| \cdot \|_2$-dense in $C[a,b]$. Is it also $\| \cdot \|_{\infty}$-dense in $C[a,b]$? In the relevant chapter of my textbook, I am given ...
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2answers
23 views

$f,g \in [0,1] , f<g $ , when is $U:=\{h \in C[0,1]:f(t)<h(t)<g(t), \forall t \in [0,1] \}$ a ball in $C[0,1]$ with respect to the sup metric

Let $f,g:[0,1] \to \mathbb R$ be continuous functions such that $f(t)<g(t),\forall t \in [0,1]$ , then under what additional conditions on $f,g$ can we conclude that $U:=\{h \in ...
0
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2answers
15 views

A question on Matrix limits

Consider $M(n, \mathbb R)$ , the set of all real square matrices of size $n$ , as an NLS with norm $||A||:=\sqrt{\sum_{i=1}^n \sum_{j=1}^n a_{ij}^2}=Trace(AA^t)$ , then is it true that a sequence of ...
2
votes
1answer
71 views

A consequence of the open mapping theorem

We let $f$ be a bounded and surjective linear map from the Banach space $X$ onto the Banach space $Y$ and put $$ r_0=\inf\{r: f(B^X(0,r)\supset B^Y(0,1)\}. $$ Using the open mapping theorem, I have ...
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0answers
35 views

Equivalence of norms in $\mathbb R^n$

Does anyone know some proofs where we use fact of equivalence of norms(I think some L continuity use this fact). Any literature of proofs would be very useful. Thank's a lot.
2
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1answer
75 views

Multiplication Operator and Supremum Norm

Let $m\in C[a,b]$. Consider on $(C[a,b], \|\cdot \|_{\infty})$ the multiplication operator $A: C[a,b] \to C[a,b], \quad Af = mf$. Prove that $\|A\| = \|m\|_{\infty}$. In my book, we are given the ...
1
vote
1answer
48 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
0
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2answers
281 views

Convex set weakly closed if and only if strongly closed as well

I'm looking for a proof that given $(X\textbf{ } \|\cdot\|)$ normed space, $M \subset X$ convex set, $M$ is weakly closed if and only if it's strongly closed as well.
3
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1answer
29 views

Norm of a functional given by difference of values

How to calculate the norm of such functional? $$\phi : \mathcal{C} ([0,1]) \ni f \rightarrow 3 f (\frac{1}{2}) - 5 f (\frac{2}{3}) \in \mathbb{K} (= \mathbb{R}, \mathbb{C})$$ If we equip ...
4
votes
2answers
55 views

Normed separable space, linearly independent $X_0 \subset X, \ \overline{linX_0} = X $

Could you tell me how to prove that a normed space $X$ is separable $\iff$ there exists an at most countable set of linearly independent vectors $X_0 \subset X$ such that $ \ \text{lin} X_0$ is ...
0
votes
0answers
19 views

What is $\sum_{i=1}^m \frac{1}{a_i^Tx-b_i}$

Ask a dumb question: We all know the following: $\sum_{i=1}^m |a_i^Tx-b_i| = ||Ax-b||_1$ How about $\sum_{i=1}^m \frac{1}{a_i^Tx-b_i}$? I think it is definitely not $ ...
1
vote
0answers
51 views

Proof that second Frechet derivative is symmetric?

Is there a "nice" way to prove that the second Frechet derivative of a function between normed spaces is symmetric? Any proofs that I've managed to find seem quite messy and don't really give any ...
1
vote
1answer
113 views

Does isomorphism between normed space implies equivalence of norm?

Let $E$ be a vector space, $\|\cdot\|_1,\|\cdot\|_2$ be two norms on $E$, if $T:(E,\|\cdot\|_1)\to (E,\|\cdot\|_2)$ be an isomorphism (linear bijection, $T,T^{-1}$ are bouned), then does this imply ...
3
votes
2answers
95 views

Good books about differentiation in normed spaces?

Typical functional analysis books don't seem to cover this subject at all, so I'm looking for some good books that deal with differentiation in normed spaces(Gateaux/Frechet derivatives etc.). ...
1
vote
1answer
81 views

Diameter of an open ball in a normed space

This is probably a silly question, but I'm reading some class notes that have the following proposition: In general it's true that $\operatorname{diam}( B(x,r) ) \leq 2r $ but in a normed space ...