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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31 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.
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1answer
50 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 ...
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1answer
47 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 ...
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2answers
85 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.
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1answer
19 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 ...
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2answers
45 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 ...
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0answers
17 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 $ ...
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30 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 ...
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1answer
76 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 ...
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2answers
66 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.). ...
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1answer
47 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 ...
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28 views

Additive function on normed linear space are linear if continuous

Is it true: Suppose $T$ is an additive function on normed linear space $X$ to a normed linear space $Y$, i.e. $T(x+y) = T(x)+T(y)$ for all $x,y\in X$ and $T$ continuous at a point. Would $T$ be ...
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0answers
19 views

Does isomorphism of duals implies isomorphism? [duplicate]

Does isomorphism (or isometric) of $X^*$ and $Y^*$ for $X,\:Y$ normed spaces (or banach) implies isomorphism (or resp. isometric) of $X$ and $Y$? I know that the other way around is true, but I never ...
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1answer
34 views

is a normed space clopen or open?

i'm having a problem with the definition of open, closed and clopen sets. I have understood the basic definitions, but then the teacher today in class said that the normed space is limitless. A ...
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1answer
36 views

Linearly independent vectors in normed vector space

Considering the sequence {$\textbf{x}_k\}_{k=1}^{N} $ in a normed vector space $X$. Assuming a constant $\alpha>0$ such that $\alpha \sum_{k=1}^{N} |c_k|^2 \leq || \sum_{k=1}^{N} c_k \textbf{x}_k ...
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1answer
31 views

Is $C_c$ dense in $L_p$ for $0<p<1$?

Let $C_c$ be the set of compactly supported functions on $\mathbb{R}$ that are infinitely differentiable. Let $S$ be the set of Schwartz functions. It is well known that $C_c$ (hence $S$) is dense in ...
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4answers
52 views

Show that $\ell^1$ is not complete with a certain metric

For $x=(x_j)_{j\in\mathbb N}\in \ell^1$ let $$\|x\|=\sup_{n\in \mathbb N}\left \Vert \sum_{j=1}^{n}x_j\right\Vert$$ Show that $(\ell^1,\|\cdot\|)$ is a normed space, but it is not complete. The ...
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0answers
26 views

Different examples in L^p spaces

I was introduced to the concept of $L^p$ spaces for the first time, and has a lot of questions on what exactly they are. If I consider the measure space $(\mathbb{R},\mathscr{B}, Leb)$, for any $p, ...
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2answers
30 views

Continuity of the multiplication map $f\mapsto x^2 f(x)$ between normed spaces

Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$. I read this solution: ...
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1answer
34 views

Additive function and continuity at a point

Does continuity at a point and Additive function imply continuity at all other points in a normed linear space. Is there some result like there exist a in field such that f(x) = ax for all x in normed ...
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2answers
76 views

Normable topology determined by its restriction to a finite number of factors?

Is it generally true that all norms $\|\cdot\|$ on a finite product of normed spaces $E_1\times\dots\times E_n$ with $\|(0,\dots,0,x,0,\dots,0)\|=\|x\|_i$ where $\|\cdot\|_i$ denotes the norm on $E_i$ ...
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1answer
31 views

On the greatest norm element of weakly compact set

Let $X$ be a Banach space and $K\subset X$ be a nonempty weakly compact set. I would like to know if there exists a point $u_0\in K$ such that $\|u_0\|\geq \|u\|$ for all $u\in K$. Thank you for all ...
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1answer
23 views

>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
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1answer
25 views

Show that the application $ Id:( C^1(0,1), \|\cdot\|_{1})\to ( C^1(0,1), \|\cdot\|_{\infty})$ is not continous

I want to prove that the application $$ Id:( C^1(0,1), \|\cdot\|_{1})\to ( C^1(0,1), \|\cdot\|_{\infty}) $$ is not continous. If I prove that this application is not bounded I have finished. So I ...
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0answers
51 views

n- dimensional normed linear space isomorphic to n-dimensional Euclidean space

If $(X,\|\cdot\|)$ is an n- dimensional normed linear space over R. Is it isomorphic to n-dimensional Euclidean space $R^n$. I know it is topologically isomorphic but what about isometry? I think if ...
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0answers
46 views

Covering Number of Lipschitz Function Space

On page 12 of the slide deck here, the author gives an example where a lower bound (and upper bound, but I am particularly interested in the lower bound) on the $\epsilon$-covering number of a ...
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1answer
43 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
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1answer
21 views

Does a normed space with norm2 defines an inner product?

I know that generally, an inner product defines a norm on an inner product space, But, generally speaking, If I have a normed space (on purpose I do not say which) with the norm 2 does it mean that I ...
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1answer
44 views

Bounded functional which composed with an unbounded operator becomes unbounded

While working on a problem, I came up with a certain lemma, however I'm not sure whether it's true and I'd be grateful for some insight. Let $ X $ and $ Y $ be normed spaces over reals, where $ X $ ...
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1answer
46 views

Isometry under condition $\|x-y\|=n\iff\|f(x)-f(y)\|=n.$ for $n\in\mathbf{N}$

Let $X, Y$ be normed spaces and $f:X\to Y$ be mapping and $n\in\mathbf{N}$ If$$\|x-y\|=n\iff\|f(x)-f(y)\|=n.$$ Under what conditions this map will be an isometry? Thanks
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66 views

Prove the triangle inequality is valid for the norm $\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$

I.e, prove $\lVert f+g \rVert\ \le \lVert f \rVert + \lVert g \rVert$ for all $f,g$ in $C^\infty [0,1]$, $$\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$$ I think we're supposed to use ...
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1answer
59 views

Prove that L 2 PC[−1, 1] is not a complete normed space

I'm trying to prove that the normed space of all piecewise continuously functions with the norm $$\int^1_{-1}|f(x)|^2dx$$ is not a complete normed space. $L_2PC[-1,1]$ for that, im trying to find a ...
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1answer
50 views

Why is $\langle x-P(x),m\rangle=0$?

Let $H$ be a Hilbert space, and let $M\le H$ be a subspace of it. Let $P:H\rightarrow M$ be the orthogonal projection $H$ onto $M$. We'll take $x\in H$, and $m \in M$. By the definition I know ...
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1answer
24 views

Why is a linear transformation of a cauchy sequence in a normed space also cauchy?

Suppose we have a cauchy sequence $\{a_n\}$ in a normed vector space $V$. Given a linear transformation $T:V \rightarrow V$, is the sequence $\{T(a_n)\}$ also cauchy? Or is it true only for finite ...
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1answer
23 views

Is the projection of a cauchy sequence in a normed finite dimensional vector space along some subspace also cauchy?

Let $V$ be a normed finite dimensional vector space. Let $S$ and $S'$ be two subspace such that $S \cap S'={0}$ and $V$ is the direct sum of $S$ and $S'$. We define the projection of a vector $x$ ...
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2answers
127 views

Is the norm of a projection of a vector along a subspace less than or equal to the norm of the vector iteself?

My question is, given a vector $x$ in a normed space, and any two subspaces with an intersection $0$ and whose direct sum is the whole vector space, is the norm of projection along one of the ...
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1answer
29 views

Is there a point that lies on the boundary of the unit ball in $\lVert\cdot\rVert_1$, and close to the zero-sequence in $\lVert\cdot\rVert_2$?

I am an engineer who is brushing up some functional analysis. I am curious about the following problem I posed to myself: Consider the sequence space of real-valued sequences that will eventually ...
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1answer
19 views

Showing a function to be a norm

I want to prove or disprove that $\parallel (x,y)\parallel=\sqrt{\frac{x^2}{9}+\frac{y^2}{4}}$ is a norm on $\mathbb{R^2}$. Since $\{(x,y):\parallel(x,y)\parallel\leq1\}$ is a convex set, ...
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2answers
195 views

Can we have something like an orthonormal basis for a finite dimensional normed space?

So I proved a certain theorem about finite dimensional inner-product spaces, but after completing the proof, I realized the only point where I used the idea of orthogonality was the construction of an ...
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2answers
50 views

Equivalence of norms on vector spaces

Let us call two norms $|x|_1$ and $|x|_2$on a finite-dimensional vector space equivalent if they set the same topology on that space. I need to show that this definition is equivalent to the existence ...
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1answer
25 views

Distance to a closed ball in a normed space.

Let $(E, \|\cdot\|)$ be a normed vector space, and consider $B = B[{\bf a},r]$ the closed ball. Let ${\bf b}\in E$. Then $\newcommand{\d}{{\rm d}} \d({\bf b},B) = 0$ if and only if ${\bf b} \in B$. ...
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1answer
34 views

Geometrical meaning of a face

Let $(X,P)$ be a locally convex space, $K$ a compact, convex subset of $X$. A face $F$ of K is a nonempty, compact, convex subset of $K$ s.t. $$\forall y,z\in K \,\forall t\in(0,1) \left[ (1-t)y + tz ...
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1answer
50 views

closure of a convex set in a normed linear space is convex ?

Is it true that if $A$ is a convex set in a normed linear space $V$ , then the closure of $A$ is also convex ? (I know that the interior is convex )
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2answers
54 views

Is $T:C[0,1]\to C[0,1]: Tf(x)=\frac12 x f(x^2)+1$ a contraction?

I am a non-mathematician (a physicist) who wants to find a continuous function on $[0,1]$ that satisfies the relation $f(x)=\frac12 x f(x^2)+1$ with $0\le x\le 1$. I need it for one of my models. ...
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3answers
160 views

Compactness of subspaces of finite-dimensional vector spaces

For subspaces of $\mathbb R^n$ we know that they are compact if and only if they are closed and bounded. Is the same true for all normed finite-dimensional vector spaces perhaps not over $\mathbb R$ ...
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2answers
61 views

A condition equivalent to orthogonality

Prove that in any inner product space: $x$ and $y$ are orthogonal if and only if $||x+\alpha y||\ge ||x||$ for every scalar $\alpha$.
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1answer
74 views

Question about a proof in Rudin's book - annihilators

I'm reading the proof of the following theorem in Rudin's "Functional Analysis": Let $M$ be a closed subspace of a Banach space $X$. The Hahn-Banach theorem extends each $m^* \in M^*$ to a ...
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1answer
67 views

How to use Triangle inequality to find the projection onto unit ball?

The projection onto the unit ball $$C:=\mathbb{B}(0,1)=\{x:||x||\leq1\}$$ is given by $$P_{C}(x)=\frac{x}{max\{||x||,1\}}, \quad\forall x\in X$$ where $X$ is Hilbert space. Now I can understand this ...
3
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2answers
74 views

How to prove a Banach normed vector space is NOT a Hilbert space?

We know that the Banach space $\big(\Bbb R^n,\|\cdot\|_2\big)$ is a Hilbert space with inner product $\langle x,y\rangle := \sum_{k=1}^n x_ky_k$. However, how to prove that $\big(\Bbb ...
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1answer
12 views

Is linear convergence norm invariant?

Let $\|\cdot\|_a,\|\cdot\|_b$ be two norms on $\Bbb R^n$ and $(x^k)_{k\in\Bbb N}\subset \Bbb R^n$ a sequence such that there exists $0<\alpha <1$ with $ \|x^{k+1}\|_a \leq \alpha \|x^k\|_a$ for ...