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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26 views

Convex subsets and Linear functionals

Let $E$ be a convex subset of a normed space $X$ and $x\in E$. Then $x\in \overline{E}$ if and only if $\Re f(x)\geq 1$ for every $f\in X'$ such that $\Re f\geq 1$ on $E$ and $\Re f(x)\leq 1$ for ...
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22 views

Give an example of a linear mapping from a normed space into a normed space which is not continuous. [closed]

Give an example of a linear mapping from a normed space into a normed space which is not continuous. I can't think of anything. Any help would be very appreciated.
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1answer
29 views

Prove that a linear mapping from a normed space into a normed space is continuous if and only if it maps bounded sets to bounded sets.

Prove that a linear mapping from a normed space into a normed space is continuous if and only if it maps bounded sets to bounded sets. I have an idea if the sets were Cauchy, but I can't assume that ...
2
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0answers
23 views

function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$ [duplicate]

I'm searching for a function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$. And it has to be $|\Omega |<\infty$. I tried $f(x)=\frac{1}{2\sqrt{x}}$ and $\Omega= ...
2
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1answer
25 views

Distance of $x$ to kernel of bounded linear functional is the norm of the functional at $x$?

Let $X$ be a Banach space and let $f\in X^*$ have norm $1$. Prove that $x\in X\implies d(x,\text{ker} f)=|f(x)|$. I have managed to prove that $d(x,\text{ker}f) \geq |f(x)|$, using a theorem that ...
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1answer
18 views

$V$ a normed space and $0_V\ne v_0 ∈ V$ . Show there exists a linear functional $θ ∈ D(V)$ satisfying $||θ|| = 1$ and $θ(v_0) = ||v_0||$ .

Let $V$ be a normed space and let $0_V \ne v_0 ∈ V$ . Show that there exists a linear functional $θ ∈ D(V)$ satisfying $||θ|| = 1$ and $θ(v_0) = ||v_0||$ . I'm not sure how to approach this. Any ...
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1answer
37 views

Application of Hahn Banach Separation theorem

I am solving an exercise (not Homework).. Let $E_1$ and $E_2$ be non empty disjoint convex subsets of $X$, with $E_1$ compact and $E_2$ closed in $X$. Then there are $f\in X'$ and $t_1,t_2$ in ...
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1answer
50 views

Open Ball and Lipschitz Equivalence equivalence

I am trying to show that two norms $\|\cdot\|$ and $\|\cdot\|^\prime$ are Lipschitz equivalent if and only if there exist numbers $r,R >0$ such that $B_r \subseteq B_1^\prime \subseteq B_R$ where ...
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0answers
51 views

completeness of $l_1 ^\infty$

I'm trying to prove that $l_p ^\infty $ is complete for each $p\geq 1$ but only with the definition of $\varepsilon$-$N$. I know that this have been proved in other posts here but I couldn't find a ...
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1answer
29 views

Definition of space $L_f^2$ where $f$ is a function?

http://it.tinypic.com/r/2iqjvbl/9 Hi guys! I'm writing my thesis for my degree and it's about Sturm-Liouville theory applications. I'm using the book "Al-Gwaiz M.A. Sturm-Liouville Theory and Its ...
1
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1answer
23 views

Convergent sequence on unit sphere

Suppose $x_n$ is a bounded sequence in a vector space $V$ with norm $||\cdot||$. Show that if: $$\hat{x}_n=\frac{x_n}{||x_n||}\;\;\text{converges}\Rightarrow x_n\;\text{has a convergent ...
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1answer
26 views

Geometry of a Cauchy sequence in a normed space

A sequence in a normed space $X$ is called a Cauchy sequence if and only if for every $\epsilon > 0$ there exists an integer $N\in \Bbb N$, such that $\|x_n-x_m\|\lt \epsilon$ for all ...
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1answer
11 views

Deducing equivalence between norms from simple condition

Let $||\cdot||,\;||\cdot||'$ be norms on $V$. Suppose for some $a,b>0$ we have: $$||x||<a\Rightarrow ||x||'<1\Rightarrow ||x||<b$$ Show that $||\cdot||,\;||\cdot||'$ are Lipschitz ...
2
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1answer
32 views

Finding closure/interior of subset of function space

Consider the subset $$A=\left\{f\in C(\Bbb R): |f(x)|< \frac{1}{1+|x|} \, \text{for all } x\in \Bbb R\right\}\subset \left\{f\in C(\Bbb R): \lim_{|x|\to \infty}f(x)=0 \right\}=X.$$ where $X$ is ...
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1answer
20 views

Do I Understand Closed Versus Complete in Metric, Normed and Inner Product Spaces?

I've looked at a number of references on this including some questions on stack exchange. Am I correct if I summarize by stating the following ? (1) A space C (metric, normed, or inner product) is ...
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2answers
44 views

Define $g :\ell_2 \to \mathbb R$ by $g(x)= \sum_{n=1}^{\infty} \frac{x_n}n$. Is $g$ continuous?

Define $g :\ell_2 \to \mathbb R$ by $$g(x)= \sum_{n=1}^{\infty} \frac{x_n}n $$ Is $g$ continuous? I need to solve this but I could not see how to tackle it? any hints or suggestion?
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0answers
78 views

Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
2
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1answer
25 views

Norms on unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+$ be its unitization. Then $||(a,z)||_1=||a||+|z|$ is a Banach algebra norm on $A^+$. Can we also make $A^+$ a Banach algebra by giving it the norm ...
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1answer
46 views

Show that $\exists$ an inner product in $X$ such that $<x,x> =||x||^2$ for all $x \in X$

Problem: Let $X$ be normed space. If on every two dimensional subspace $Y$ of $X$, there is an inner product $<,>_Y$ such that $<y,y>_Y=||y||^2$ for all $y\in Y$. Then there is an inner ...
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1answer
55 views

Confused by peculiar norm

Let $X$ be an infinite subset of $ [0,1]$. In an exercise I am considering the norm on $P([0,1])$ (polynomials on unit interval) defined by: $$||p||_X=\sup_X |p|$$ My question is, how do I make sense ...
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1answer
33 views

Completness of vector spaces and equivalent norms. [closed]

If a vector space V is complete in a norm ||.||, then is complete in any norm ||.||' equivalent to ||.||.
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142 views

Uncountably many norms such that no two are Lipschitz equivalent

I am struggling with the following question: Is it possible to find uncountably many norms on $C[0,1]$ such that no two are Lipschitz equivalent? I had thought about trying to define norms for each ...
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0answers
28 views

Proving that given this property then the norm is induced by a inner product

Let $(X,||\cdot||)$ be a normed space such that, for $x,y\in X$ $$||x + y||^2 +||x - y||^2 = 2||x||^2 + 2||y||^2$$. Then I want to check that $||\cdot||$ is induced by an inner product, so what I ...
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3answers
59 views

Why aren't the rationals a compact subset of $\mathbb{R}$?

We define a compact subset of some normed vector space $V$ to be any subset $S$ where every sequence $\{\mathbf{x}_{n}\}$ in $S$ has a subsequence which converges to some $\mathbf{x}$ in $S$. Then ...
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1answer
27 views

If $ T(f)=\int_0^12xf(x)\,dx$ find the value of $||T||$

Let , $T:(C[0,1],||.||_{\infty})\to \mathbb R $ be defined by $\displaystyle T(f)=\int_0^12xf(x)\,dx$ for all $f\in C[0,1]$. Then find $||T||$, where , $\displaystyle ||f||_{\infty}=\sup_{0\le ...
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2answers
31 views

Separability of $l_p(I,K)$

Good day, I have the following question "Prove that for $1 \leq p < \infty$, $l_p(I,K)$ is separable if and only if $I$ is countable, and $l_{\infty}(I,K)$ is separable if and only if $I$ is ...
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1answer
67 views

On continuous mappings on closed unit balls which is injective in the interior

Let $f: B[\theta,1] \to B[\theta , 1]$ be continuous and is injective in $B(\theta , 1)$ ; then is it true that the set $\{x \in Bd \space B(\theta,1): |f^{-1}(\{x\})|\ge3\}$ is countable ? (here ...
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2answers
26 views

How does one prove that two norms are equal if and only if their closed 1-balls are equal?

Let $X$ be a vector space and let $||\bullet||_1$ and $||\bullet||_2$ be two norms on $X$. I wish to prove that $||x||_1=||x||_2$ for all $x\in X$ if and only if $\{ x\in X \text{ such that ...
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0answers
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Continuous linear bijection of a Banach space is a homeomorphism

I have seen an example of a continuous linear bijection $f:S\to S$, where $S$ was a normed linear space, such that the inverse function $f^{-1}$ was not continuous,as it was unbounded.The norm on $S$ ...
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3answers
86 views

$T$ is surjective if and only if the adjoint $T^*$ is an isomorphism (onto its image)

I am trying to prove the following statements: Let $X$ and $Y$ be normed spaces (not necessarily complete) Let $T\in L(X,Y)$ (meaning $T:X\to Y$ is a bounded linear map). Let $T^*:Y^*\to X^*$ denote ...
0
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1answer
36 views

Is there any continuous function $f:D^n \to S^{n-1}$ whose restriction to the sphere is the identity?

Is there any continuous function $f:D^n \to S^{n-1}$ whose restriction to the sphere is the identity ? If there does not exist such a function then can we deduce Brouwer fixed point thoerem from this ...
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0answers
67 views

On the form of injective function $f:\mathbb R^+ \to \mathbb R^+$ such that $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ is a norm

Let $f:\mathbb R^+ \to \mathbb R^+$ be injective such that on $\mathbb R^n , n>1)$ , $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ , where $u=(u_1,u_,\ldots,u_n)$ , defines a norm ; then is it true ...
3
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1answer
43 views

Is every closed set , the set of zeroes (resp.critical points) of some smooth real valued function? [duplicate]

Let $A$ be a closed subset of $\mathbb R^n$ : 1) Is it true that for some smooth function $f: \mathbb R^n \to \mathbb R$ , $A=f^{-1}(\{0\})$ 2)Is it true that for some smooth function $f: \mathbb ...
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0answers
25 views

What does a norm which is symmetric “around” an index subset look like?

I am looking at norms $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ that have the following symmetry property $$\forall \beta \in \mathbb{R}^{n} \text{: }, \lVert\beta \rVert=\lVert ...
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2answers
42 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} = ...
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1answer
28 views

dual pairs in $\mathbb R^n$ with non-standard norms $l_1$ and $l_{\infty}$

Suppose $X=Y=\mathbb R^n$. Usually, to apply a separating hyperplane theorem in $X$ (the primal space), we associate both $X$ and $Y$ with the standard Euclidean norm. My question is that could I ...
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1answer
26 views

Restrictions of function decomposition in $R^3$

I'm interested in the properties of countable basis functions that span functions living in $\Bbb R^3$. Can I represent a $L^2$ normalizable function that has a point divergence, (for example, ...
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1answer
40 views

Is it possible to strengthen this inequality?

Let $T:X\to Y$ be a linear operator from a normed space $X$ into a normed space $Y$. Suppose that $T$ has the property that for a fixed $y\in Y$ and any $\alpha>1$, there exists an $x_{\alpha}\in ...
0
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1answer
32 views

Composition of a function and a scalar

Are the following true? $\operatorname{scalar} \circ \operatorname{function} = \operatorname{scalar} \times \operatorname{function}$ $\operatorname{function} \circ \operatorname{scalar} = $ the ...
1
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1answer
32 views

X is Banach iff $\sum_{n \geq 1} y_n$ converges, where $\left \| y_n \right \| \leq 2^{-n}, \forall n$

Prove that the normed space $X$ is Banach space if and only if $\sum_{n \geq 1} y_n$ converges, where $\left \| y_n \right \| \leq 2^{-n}$ for all $n$.
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1answer
38 views

Prove by Induction that Norms in a Finite Dimensional Space are Equivalent?

I would have thought that this is a good candidate for an inductive proof, but I have searched for one and failed. Is there such a proof, and if not why not ? Here's how far I got. It's easy to ...
6
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1answer
74 views

Reflexive Banach space: Boundedness of subset implies weak compactness. Closed or not?

Claim:In a reflexive Banach space, the weak compactness of a subset is equivalent to the boundedness of the subset. But there is no guarantee that the bounded subset would even have its sequences ...
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1answer
16 views

Finding a compact set containing the unit ball in a normed space

I would like to show that there is a compact set $K \supset \{ \Vert x \Vert \leq 1 \}$ in a general normed vector-space $X$, but I have no clue how to do it. Or is it maybe possible to have a finite ...
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0answers
20 views

Completeness of the space $C_{2\pi} \left ( \mathbb{R} \right )$ [duplicate]

Denote $C_{2\pi} \left ( \mathbb{R} \right )$ is the set of complex-value, continuous on $\mathbb{R}$, periodic functions with period $2\pi$. Fix $p \in \left [1, \infty \right )$. For each $x \in ...
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0answers
28 views

Invertible linear transformations between a set with infinity norm and euclidean norm

Let $ n \geq 2$. Show that there is no invertible linear transformation between $ S^1 := \{ x:\|x\|_{\infty} = 1\}$ and $S^2 := \{ x : \|x\|_2 =1\} $ as subsets of $( \mathbb{C}^n, \|.\|_{\infty}) $ ...
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2answers
25 views

Norms are not equivalent in $c_0$

Consider two norms in the space $c_0$: $$\lVert x \rVert = \sup \lvert x_i \rvert$$ and $$\lVert x \rVert _0=\sum 2^{-i} \lvert x_i \rvert$$ Prove that above two norms are not equivalent. I know ...
3
votes
3answers
61 views

What does $\mathbb{R}^\mathbb{N}$ actually mean?

We all know what $\mathbb{R}^n$ means. But I came across this statement about $\mathbb{R}^\mathbb{N}$ in a note that says $\mathbb{R}^\mathbb{N}$ is a vector space under pointwise operations has no ...
2
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0answers
66 views

Prove that X is a Banach space.

I have this problem to solve and I am not sure if I'm doing it right. Let $X$ denote a real vector space consisting of all continuous functions $u \colon[0, \alpha] \to \mathbb R$ such that their ...
0
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1answer
25 views

Proof of the continuity method, guidance

Let $\mathcal{B}$ be a Banach space, and $V$ a normed linear space. $L_0,L_1:\mathcal{B}\to V$ are bounded linear operators. Assume $\exists c$ such that $L_t := (1-t)L_0 + tL_1$ satisfies: ...
0
votes
1answer
18 views

$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$ Why is $A$ injective?

Why does $A:X\to Y$ where $X$ is a banach space and $Y$ is a normed space, where $A$ is a surjective bounded linear operator, where: $$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$$ Mean that $A$ is also an ...