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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30 views

Determine whether a sequence space is normed

Let $(a_k)$ be a monotonously decreasing sequence of positive numbers with $a_1 =1$ and $a_k\to 0$. Let also $\sum_{k=1}^\infty a_k$ diverge. If $1\leq p<\infty$, show that $$A := \lbrace (x_k) : ...
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2answers
38 views

Is the dual space of a separable normed space also separable?

If $X$ is a real normed space such that $X^*$ (the dual) is separable then $X$ is also separable. Is the converse true, i.e., if $X$ is separable then is its dual space $X^*$ necessarily separable?
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45 views

Does every finite dimensional subspace of any normed linear space have a closed linear complement?

Let $Y$ be a finite dimensional subspace of a real NLS $X$ and let $Y=span \{y_1,...,y_n\} $ ; then by Hahn-Banach theorem , we can find continuous linear functionals $l_1,...,l_n \in X^*$ such that ...
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2answers
64 views

Proof of a p-norm and Cauchy/Cauchy-Schwarz inequality generalization

I've started to study metric spaces recently and got intrigued with a demonstration which led me to a bunch of ideas and questions. First of all, I've proved the Cauchy inequality and then ...
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1answer
39 views

Convergent series which is not absolutely convergent in $(X,\|\cdot\|)$

Let $(X,\|\cdot\|)$ be a normed linear space. Recall from prior results that $(X,\|\cdot\|)$ is Banach $\iff$ any absolutely convergent series in $(X,\|\cdot\|)$ converges. (a) Give an example of ...
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1answer
36 views

Functional Analysis(Normed Spaces)

Let $X$ be the space of all complex valued square Riemann- integrable functions on $[0,1]$ with $2$- norm. Define the map $F:X\to X$ by $F(u)=v$ with $v(t)=\int\limits_{0}^{t}{ u^2(s) ds}$, then ...
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149 views

Regarding linear independence on a normed-linear space given a condition

Let $(X,\|\cdot\|)$ be a normed linear space and $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ linearly independent vectors in $X$. Show that there exists $\epsilon > 0$ such that if $y_{1}, y_{2}, ...
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1answer
45 views

A question in the proof of $C(X)$ is not a dual space of a Banach space.

Let $X$ be a non-singleton compact connected space.I want to show that $C_{\mathbb{R}}(X)$ is not the dual space of a Banach space,$\mathbb{R}$ is real number field. I already know that, extreme ...
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1answer
56 views

Prove that there exists NO norm such that $ f_n $ converges to $f$ iff $f_n$ converges to $f$ on compacta.

More specifically: Given the space of continuous functions $\mathbb{R}\rightarrow\mathbb{R}$ where convergence is equivalent to uniform convergence on compacta (i.e., compact sets), prove that ...
2
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1answer
224 views

Regulated function-higher dimensions

Let $\mathbb{X}$ be a Banach space and let the function $f:[0,1]^2 \rightarrow \mathbb{X}$ be continuous. (a) Show that $\forall\ \epsilon>0\ \exists\ \delta>0$ s.t. ...
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39 views

How to show that $BV[0,1]$ ( the space of all functions on $[0,1]$ of bounded variation ) is not complete under supremum norm?

How to show that $BV[0,1]$ ( the set of all functions of bounded variation ) is not complete under supremum norm , by explicitly constructing a Cauchy sequence which does not converge or by ...
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2answers
52 views

Normed vector space with a closed subspace

Suppose that $X$ is a normed vector space and that $M$ is a closed subspace of $X$ with $M\neq X$. Show that there is an $x\in X$ with $x\neq 0$ and $$\inf_{y\in M}\lVert x - y\rVert \geq ...
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1answer
22 views

Inequality problem in normed space involving maximum

Let $X$ be a normed space. Show that for every $x,y\in X$: $$\|x+y\|\leq \max\{\|x\|,\|x+2y\|\} $$ Wanted to check two cases. First, assume the maximum is $\|x+2y\|$, then $$\|x+y\|\leq ...
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1answer
65 views

Minimizing a funtional in the Sobolev space $H_0^1$

I am trying to show that, given $f \in H^{-1}(U)$, there exists a unique $u \in H_0^1(U)$ such that: $$\int_U \nabla u\cdot\nabla v \, \mathrm{d}x= \langle f,v \rangle_{H^{-1}} \, , \quad \forall \, v ...
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3answers
28 views

Elements of a space

If A is a normed linear space and there exists an element x,y in A, then is it also true that x-y is also in A as well? I am pretty sure this is true by definition, but I am not able to find any ...
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1answer
61 views

integral equation, bounded, norm [on hold]

to be honest i'm not too sure where to start? i'm assuming I have to estimate the norm in the hint using the supremum norm in order to estimate $k(s,t)$ and then somehow use this information to ...
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2answers
36 views

When we raise $f$ to a positive power, what happens to the norm?

I am missing something in the identity $(1.18)$ below. What does that identity have to do with the fact that the $L^p$ norm is a non-negative number? The identity $(1.18)$ says that when we raise ...
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0answers
20 views

Applying homogeneity for a norm to prove triangle inequality

I am hoping to have someone help explain to me the steps of this proof of the triangle inequality for $L^p$ spaces, $p\ge 1$. Specifically the first two applications of the homogeneity property don't ...
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0answers
37 views

When is a topology is defined for this special kind of uniform convergence?

Question: Let $\phi:X\to X$ be bijective and continuous. Does a topology $\tau$ exist such that $$f_n\overset\tau\to f\Leftrightarrow \phi\circ f_n\overset\infty\to\phi\circ f$$ where $\infty$ ...
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1answer
67 views

If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$?

$U$ is a bounded open subset of $R^n$. If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$ ?
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0answers
16 views

Introduction to a textbook on Minkowki spaces

I want to learn more about the metric spaces, specially the "Minkowski spaces" and "Zermelo navigation problem" on Minkowski spaces. I have just studying the book " Riemann-Finsler Geometry" by ...
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2answers
93 views

showing a space with sup norm is complete

show that $(c_0,||\cdot||_\infty)$ be the space of real valued sequences converging to $0$ with the supremum norm is complete Okay so I know that the normed space is complete if any Cauchy sequence ...
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0answers
38 views

Triangle inequality in norms

Let $N:V\to\mathbb{R}$ be a sequence of norms and $d(x,y)=\sum_{j=1}^{\infty}2^{-j}\frac{N_{j}(y-x)}{1+N_j(x)+N_j(y)}$ Show that $d(x,z)\leq{d(x,y)}+d(y,z)$ I tried the following ...
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1answer
35 views

$\sim$ is an equivalence relation on the set of norms on $X$.

Let $X$ be a vetor space. Prove that $\sim$ is an equivalence relation on the set of norms on $X$. Where $\sim $ is the equivalence of two norms. This seems very abstract. What exactly do I have to ...
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1answer
61 views

bounded functions, norms

Instead of answering this question could someone possibly explain what I need to do? I don't fully understand what the question is asking, firstly $M$ has been fully defined so how do we know $|M|$? ...
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1answer
65 views

Prove $\|\cdot\|_p$ and $\|\cdot \|_q$ aren't equivalent on $\ell^p$

$1 \leq p < q < \infty$ So we need to find something in $\ell ^p$ that gives different results in each of the two norms but I can't think of anything. I could think of something that is n ...
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1answer
38 views

Prove $\| \cdot \|_{\infty}$ is well-defined on $\ell ^1$

$1 \leq p < q \leq \infty$ $p=1$ and $q= \infty$ Let $(a_k)_{k \geq 1} \in \ell^1$, then $$\sum_{k \geq 1}|a_k|^1< \infty$$ (*) which means $\sup_{k \geq 1 } |a_k|^1 < \infty$. Hence it is ...
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1answer
34 views

$X,Y$ be normed-linear spaces , $Y$ finite dimensional , $T: X \to Y$ be a non-continuous linear map , then is $\ker T$ dense in $X$?

Let $X,Y$ be normed-linear spaces ( over $\mathbb R$) , $Y$ is finite dimensional and $T: X \to Y$ be a linear map which is not continuous ; I know that then $\ker T$ is not closed in $X$ ; my ...
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1answer
49 views

Prove $\sum \frac{\sin(n^2 t)}{n^2}$ converges

Let $(X,\|\cdot\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+\cdots+x_n$ ...
2
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2answers
48 views

Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is ...
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1answer
30 views

Prove $\sup _{t \in [0,1]} |P(t)|$ is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P \in X$, define $N_1 (P)=\sup _{t \in [0,1]} |P(t)|$ I am having trouble proving the part when you show ...
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1answer
44 views

An open ball is an open set by the continuity of the norm

I Hvev an exercise that I should prove that in a normed space, an open ball is an open set, but using the property of the continuity of the norm. Given a normed space $X$ and the norm $||.||$ such as ...
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0answers
15 views

Norms on matrices over Banach algebras

Let $A$ be a unital Banach algebra. Is there a sequence of norms $||\cdot||_n$ such that $(M_n(A),||\cdot||_n)$ is a Banach algebra for each $n$, identity matrices of all sizes have norm 1, the ...
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1answer
69 views

$X,Y$ infinite dimensional NLS , not both Banach , then $\exists T \in \mathcal L(X,Y)$ such that $R(T)$ is not closed in $Y$?

Let $X,Y$ be infinite dimensional normed-linear spaces , not both Banach , then does there necessarily exist a continuous linear transformation $T:X \to Y $ such that $range (T)$ is not closed in $Y$ ...
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1answer
52 views

Alternative formulation of the supremum norm?

The supremum norm is defined as $$\|f\|_\infty=\sup\limits_X|f|$$ This induces a topology: $$ \begin{align}f_n\overset\infty\to g&\Leftrightarrow \sup\limits_X|f_n-g|\overset{\mathbb{R}}\to 0\\ ...
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1answer
58 views

Sequence of partial sums converge

Let $(X,\|.\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+...+x_n$ converges ...
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3answers
355 views

What points is the norm Frechet differentiable at

I know the definition of Frechet derivatives - there exists a bounded linear map... Maybe someone could show me a similar example on how to approach questions like these.
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1answer
51 views

Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a Banach space for any norm $\|\cdot\|$ on it?

Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a complete space for any norm $\|\cdot\|$ on it ?
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1answer
40 views

Proving these norms are not equivalent

$X$ is the vector space containting all polynomials with real coefficients. For every $P \in X$, define $N_1(P)= \sup _ {t \in [0,1]} |P(t)|$ and $N(P)=N_1(P)+|P'(1)|$. Prove that $N$ is not ...
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1answer
68 views

Can any uncountable dimensional real vector space be made into a Banach space?

On any real vector space $V$ of uncountable dimension , can we always define a norm such that endowed with that norm , $V$ becomes a complete normed linear space ? ( I know it can be done if $V$ is ...
2
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1answer
61 views

$X$ be a real normed linear space ; if $\mathcal L(X,X)$ is complete then is $X$ also complete?

Let $X$ be a real normed linear space and $\mathcal L(X,X)$ denote the set of all bounded linear operators on $X$ , we know that if $X$ is complete then so is $\mathcal L(X,X)$ ; is the converse true ...
2
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1answer
80 views

Lipschitz map between metric and normed spaces

Let be $F:(X,d)\to V$ a map between $(X,d)$ metric space and $V$ normed space, such that for each $f\in V'$ (linear and continuous), $f\circ F$ is lipschitz map. Show that $F$ is a Lipschitz map. I ...
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0answers
44 views

How to see if a norm is finite

How do you know if $$||\frac{1}{\sqrt3},\frac{1}{\sqrt8}, ... , \frac{1}{\sqrt{n^2-1}}, ... ||_2$$ is finite. So this means is $$\bigg( \sum _{n=2}^{\infty} | \frac{1}{\sqrt{n^2-1}}|^2 \bigg) ...
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0answers
6 views

Two points in a polygonal-path-connected set can be connected with a non-intersecting polygonal path

Let $X$ be polygonal-path-connected and $x,y\in X$. So $x$ and $y$ can be connected by a polygonal path $P=\bigcup_{i=1}^n L_i$ where $L_i$ is a line segment $[x_i,x_{i+1}]$. Non-intersecting means ...
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1answer
36 views

Show that $ f(x)=\sum_{n=1}^{\infty} 2^{-n} f_n(x)$ defines a continuous function on $(0,\infty)$

Let $f_n$ be a sequence of continuous functions on $(0,\infty)$ with $|f_n(x)|\le n$ for every $ x>0$ and $n\ge1$, and such that $\lim_{x\to\infty} f_n(x) =0$ for each $n$.Show that $ ...
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1answer
352 views

Frechet Derivatives of normed spaces

(a) Would I use the definition of an open set for one U? How do I show the function is Frechet differentiable. I know the definition but not sure how to apply it. $\lim_{h\to 0}\frac{\lVert ...
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1answer
44 views

Proving polynomial v.s. is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P ∈ X$, define $N_1(P) = \sup_{t∈[0,1]} |P(t)|$ and $N(P) = N_1(P) + |P'(1)|$. I have to prove $N_1$ is a norm ...
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1answer
569 views

For every normed space the norm map is not Fréchet differentiable at $0$.

Argue that for every normed space $\mathbb{X} \neq \{ 0 \}$ the norm map $\| \ldotp \|_\mathbb{X} : \mathbb{X} \to \mathbb{R}$ is not Fréchet differentiable at $0$. Not really sure where to start ...
0
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2answers
53 views

Prove that $C^1[0,1]$ is space of continuously differentaible function with $C_1$ norm is separable.

$C^1[0,1]$ is space of continuously differentiable function with $C_1$ norm.Then the space $ (C^1[0, 1],)$ is a separable space. I am thinking of c^1[0,1] is subset of c[0,1], and c[0,1] is separable. ...
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1answer
55 views

Proving something is a Banach Space

Prove that $(\ell ^∞,||·||_∞)$ is a Banach space using the following steps. Let $(x_n)_{n∈\mathbb N}$ be a Cauchy sequence in $(\ell ^∞,||·||_∞)$. For $n > 1$, let $x_n = ...