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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98 views

Normed vector spaces inequalities: proving by contradiction

Often when there is some inequality that we want to prove in a normed space, the proof goes something like "Assume there's a sequence $f_n$ with $\lVert f_n \rVert = 1$..." Would someone give me a ...
2
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3answers
254 views

Boundedness and pointwise convergence imply weak convergence in $\ell^p$

Let $p\in(1,+\infty)$ and consider the space $\ell^p$ with its usual norm. The following are equivalent: (1) $x_n \rightharpoonup x$ (i.e. $x_n$ weakly converges to $x$); (2) $$\exists ...
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1answer
193 views

Equivalence between $Lip \ norm$ and $C_1 \ norm$.

Let $f\in C^1([a,b])$. Prove that $\|f\|_{C^1} = \|f\|_{Lip}$. By definition of Lip norm and $C^1$ norm, it is equivalent to prove that $\|f'\|_{\infty}=Lip(f,(a,b))$, where the second member is the ...
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1answer
591 views

Weakly sequentially compact sets

From Peter Lax Functional analysis page 104: Show that a weakly sequentially compact set is bounded. Definition. A subset $C$ of a Banach space $X$ is called weakly sequentially compact if ...
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3answers
152 views

Show that a linear functional does not belong to a dual space.

So I have the following statement to prove Let $L:C\:[0,1]\to \mathbb{C}$ be a linear functional defined by $$ Lf=f(0)$$ Show that $L\notin(C[0,1],||\cdot||_2)^*$, where $||\cdot||_2$ is the ...
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1answer
102 views

Proper Linear Subspaces

Let $X$ be a normed space. I want to prove that for any proper linear subspace $M$, there exists a point $x$ with $||x||=1$ such that $inf\{||x-y||:y\in M\}>1-\epsilon$ for arbitrary epsilon. ...
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0answers
67 views

Does $X^\ast$ separable imply $X$ separable? [duplicate]

Possible Duplicate: Proof: $X^\ast$ separable $\implies X$ separable Suppose $X$ is a normed vector space. Does $X^\ast$ separable imply $X$ separable? If $X$ is complete, the answer ...
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1answer
448 views

proving $L^\infty$ norm inequality (disprove $\Vert f\Vert_\infty\le\sqrt{n}$)

There are three parts in this question, I've done the first two but not sure about the third one. Also see $L^2$ norm inequality. In the third part, I am asked to show that if $W$ is a ...
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3answers
194 views

A question about the proof of the open mapping theorem in “Functional Analysis” by Rudin

In the proof of the open mapping theorem in "Functional Analysis" by Rudin, there is the following argument: Let $X$ be a topological vector space in which its topology is induced by a complete ...
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3answers
847 views

Show that the unit sphere is strictly convex

I can prove with the triangle inequality that the unit sphere in $R^n$ is convex, but how to show that it is strictly convex?
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1answer
67 views

find the Fréchet derivative of $F : [0,1] \times \mathcal{C}([0,1]) \rightarrow R : (x,f) \mapsto f(x)$, in $(x_0,f_0)$

As an exercise for my analysis class, I have to find the Fréchet derivative of $F : [0,1] \times \mathcal{C}([0,1]) \rightarrow R : (x,f) \mapsto f(x)$, in $(x_0,f_0)$, where $f_0$ is differentiable ...
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1answer
288 views

The openness of the set of positive definite square matrices

Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries. For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by: $$ \displaystyle\|A\|_1=\max_{1\leq j\leq ...
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2answers
158 views

On $L^p$ and $\ell^p$

If a continuous and infinitely differentiable function $f(x): \mathbb{R}\to\mathbb{C}$ is in $L^p$, is it also true that $f(n),\ n\in \mathbb{Z}$ is in $\ell^p$?
2
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1answer
146 views

Why is $L^3$ weaker than $L^2$?

Someone told me today that if I can show $\Vert A_n-B_n\Vert_3\to 0$ as $n\to \infty$, then claiming $A=B$ as $n\to \infty$ (where $A$ and $B$ are the respective limits of $A_n$ and $B_n$) is a weaker ...
14
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1answer
506 views

Multiplicative norm on $\mathbb{R}[X]$.

How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$. ...
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1answer
46 views

Continuity of $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$

I have a question about a proof in my analysis textbook. They show that if $E$is a banach space, then $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$ is continuous by first showing that it is ...
2
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1answer
148 views

Dual space norms and equivalence

Suppose $S(r)$ is set parametrised by $r \in [0,T]$. Let $\phi_t^s : H^1(S(t)) \to H^1(S(s))$ is a linear homeomorphism. Suppose $\lVert \cdot \rVert_{H^1(S(t))}$ and $\lVert \phi_t^s(\cdot) ...
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0answers
130 views

Separation in infinite dimensional normed space

I would like to construct some counterexamples: $E$ is an infinite dimensional normed space. Let $C, D$ be nonempty convex subsets in $E$ such that $$ C\cap D=\emptyset. $$ There is no vector $f\in ...
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2answers
183 views

Hilbert dual space (inequality and reflexivity)

Let $V \subset H$ where $H$ is Hilbert space. Let $T:H^* \to V^*$ be the canonical map that restricts the domain of a functional in $H$ so that it's a functional in $V$. How do I show that $$\lVert ...
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3answers
369 views

The role of dual space of a normed space in functional analysis

We have known that dual space of a normed space is very important in functional analysis. I would like to ask two questions related dual space of a normed space: What is the motivation of ...
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1answer
3k views

Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
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2answers
91 views

functional analysis-normed linear space

Can somebody please help me to find the answer for this problem... Let $V$ be a norm linear space and let $x\in V\setminus\{0\}$. Also let $W$ be a linear subspace of $V$. Show that if there is ...
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1answer
2k views

Let $X$ be an infinite dimensional Banach space. Prove that every basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
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1answer
289 views

Space Completion Theorem for Normed Spaces

I went to a functional analysis course this year and my lecturer wrote down this Theorem. Lots of students pointed out it is incorrect, but she insisted it was. I am stating it now and hope someone ...
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2answers
421 views

Continuity in a normed space

Let $X$ be a normed space. Show that the function $f:X \to R$ defined by $f(x)=\|x\|$ is continuous on $X$.
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0answers
74 views

some facts about $L_p$ space

I was wondering relations of $L_p$ spaces.. Let $E$ be a measurable set. If $E$ is of finite measure, then $L_p(E) \subset L_q(E)$, $1 \le p \le q \le \infty$. However, does it still hold if $E$ is ...
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4answers
305 views

Sum of closed convex set and unit ball in normed space

Let $(X, \|.\|)$be a real normed space. Let $A$ be a closed convex suset of $X$ and $\mathbb{B}$ a unit ball in X, i.e. $$ \mathbb{B}=\{x\in X: \|x\|\leq 1\}. $$ I would like to ask whether ...
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1answer
138 views

Sum of weak star closed set and compact weak star set

Let $X$ be a real Banach space and $X^*$ be its dual space. Let $C$ be a weak$^*$ closed subset in $X^*$ and $D$ a compact weak$^*$ in $X^*$. I would like to ask whether $C+D$ is closed weak$^*$ in ...
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1answer
78 views

A normed space embedding question

Does any normed space X can be embedded into another normed space Y, such that X is density in the Y and dim(Y)=dim(X)+1.
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2answers
592 views

Series in incomplete normed space

We have known that "A normed space $X$ is a Banach space if and only if each absolutely convergent series in X converges". We would like to find an explicitly incomplete normed space and an explicitly ...
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2answers
123 views

My “wrong” comparison between $\ell^2$ and $\ell^1$

For sure, $\ell^2$ is larger than $\ell^1$, because for $|x|<1$, $|x|^2<|x|,$ that is, $||x||_2\leq||x||_1.$ But using Cauchy-Schwartz inequality, I get a "wrong" comparison: ...
2
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2answers
122 views

on proving that $\|\cdot\|_2$ is a norm on $C[0,1]$

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$ and consider the vector space $C[0,1]$, the collection of continuous functions $f\colon[0,1]\to\mathbb{F}$. I want to show that $\|\cdot\|_2$ is ...
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1answer
121 views

convergence in function space

Maybe is a silly question, but for some reason I am confused... If $\mathcal{F}$ is a normed space of real functions and $\displaystyle{ f \in \mathcal{\bar F } }$ then there exists a sequence of ...
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1answer
44 views

integrable, $L_1$ and $L_\infty$

I have a question about normed space and integrable. If $f$ is in $L_\infty$, $g$, which is $g \le f$, can be absolutely integrable ($g$ is in $L_1$)? And how can I prove it?
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1answer
121 views

Convergence in $L_\infty$ and $L_1$ even if infinite measure space

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. In the literature, assuming the measure space $X$ has finite measure, if $f_n$ converges to ...
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2answers
172 views

Intersection of a unit sphere of a given norm in finite dimension with an hyperplane.

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$. Let $C:=\{x\in\mathbb{R}^n\,:\,\|x\| \leq 1\}$, that is to say let $C$ be a convex compact symmetric set of non empty interior. Let $H$ be a linear ...
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5answers
192 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 $X$ ...
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2answers
893 views

Is closure of linear subspace of X is again a linear subspace of X??

Let $X$ be a normed linear space with norm $||\cdot||$ and $A \neq \emptyset$ is a linear subspace of $X$. Prove that $\bar{A}$ is also a linear subspace of $X$.
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2answers
452 views

Show that the discrete metric can not be obtained from $X\neq\{0\}$

If $X \neq \{ 0\}$ is a vector space. How does one go about showing that the discrete metric on $X$ cannot be obtained from any norm on $X$? I know this is because $0$ does not lie in $X$, but I am ...
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1answer
352 views

Finding a cauchy sequence that does not converge on M

We define the following infinity norm on $\mathbb{R}$ as follows $$l_\infty(\mathbb{R}) = \{ (x_i)_{i \in \mathbb{N}} \,\mid\, x_i \in \mathbb{R}, \sup_{i\in\mathbb{N}} \left|x_i\right|<\infty \}$$ ...
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1answer
38 views

Replacing one of the conditions of a norm

Consider the definition of a norm on a real vector space X. I want to show that replacing the condition $\|x\| = 0 \Leftrightarrow x = 0\quad$ with $\quad\|x\| = 0 \Rightarrow x = 0$ does not alter ...
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1answer
119 views

Question about norms and coarseness of topology

I've been thinking about norms and asked myself the following question: If I have two norms $\|\cdot\|_A$ and $\|\cdot\|_B$ with $\|\cdot\|_A \leq \|\cdot\|_B$, which topology is coarser, that is, ...
5
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1answer
638 views

Equivalence of reflexive and weakly compact

In a normed space $X$ is there an equivalence between these two proposition? $1)$ $X$ is reflexive; $2)$ $B$, the unit ball of $X$, is weakly compact.
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2answers
1k views

Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$

Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on ...
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1answer
358 views

Distance between real finite dimensional linear subspaces

Is there a usual distance between linear subspaces ($V,W$) of an n-dimensional normed vector space with inner product? In the case of hyper-planes one could use the angle (based on the inner product ...
2
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1answer
244 views

$C^1 [0,1]$ with different norm

If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where $$ \Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)| $$ for any $f\in C^1 [0,1]$, is this space Banach? ...
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0answers
210 views

equivalence of norms in an open set

I would like to prove that two given norms in the space of smooth functions are equivalent in an open set, is it enough to show that they are equivalent for any compactly contained open set? why? ...
2
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1answer
219 views

inequality between norms of vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. It is known that $\|a\|_{\infty}\leq \|a\|_2\leq \sqrt n\|a\|_{\infty}$. Let $k<<n$. For which kind of vectors the following would be true: $$ ...
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3answers
184 views

If $(x_n)$ converges weakly to $x$, then $x$ is in the closure of the span of the $x_n$

I need your help with this problem that I founded it in a lecture notes. Then, the problem says: Let $ X $ be a normed space. Show that if a sequence $ (x_n) _ {n \in \mathbb {N}} $ in $ X $ ...
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3answers
184 views

How to show convexity of a ball in metric space?

If $(X,\|\cdot\|)$ is a normed linear space, then how to show any ball $B(x,r)$ is convex? I know that if $x,y\in A\subset V$ then $[x,y]\subset A$, where $A$ is a convex subset of vector space $V$ ...