A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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Compactness of the closed unit ball of ${\rm Lip}_0(X)$ in the topology of pointwise convergence implies that ${\rm Lip}_0(X)$ is a dual space

It is proven that a closed unit ball in the set of real-valued Lipschitz functions ${\rm Lip}_0(X)$ defined on a Banach space $X$ is compact for the topology of pointwise convergence. However, I ...
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36 views

Intersection of decreasing sequence of non empty closed sets

Let X is Banach space. Suppose $B_n$ are decreasing sequence of non empty closed balls. Prove their intersection is non-empty. I have some idea. Idea is pick $x_n \in B_n\backslash B_{n-1}$ in such ...
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13 views

Integration w.r.t nondegenerate Gaussian probability measure on $X$ with mean $0$

Suppose that $X$ is a Banach space. Denote $\gamma$ as a nondegenerate Gaussian probability measure on $X$ with mean $0$. Question: Is it true that $$\int_X{d\gamma(t)}=0?$$ Or we have ...
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43 views

Functor in $\mathbf{Ban}$ that puts exact sequences into exact sequences

Let $F$ - is functor in category of Banach spaces $\mathbf{Ban}$ with follow property: $f_n : A_n \to A_{n+1}$ exact sequence iff $F f_n : F A_n \to F A_{n+1}$ is exact sequence. Is it true, that $F$ ...
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37 views

In a Topological vector space, a subspace of codimension 1 is either dense or closed.

Let $X$ be a topological vector space and $V$ be a linear subspace of $X$ such that $\text{dim}(X/V)=1$, then either V is closed or $\overline{V}=X$. In other words if it is not closed then it ...
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1answer
74 views

Does there exist a Banach space with no complemented closed subspaces?

I know that every Hilbert space can be decomposed as the direct sum of two non-trivial closed subspaces, eg. taking the kernel and range of any non-trivial bounded projection. But I don't know what ...
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76 views

Universal property of l^p-spaces

The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product: Let ...
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32 views

Existence of a linear map from a dense space

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
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32 views

Show that any convex subset of a Banach space X is closed with respect to the norm if and only if it is closed in the weak topology

How can it be proved? In a way which could be understood by a undergraduate math student. Specially if and "only if".
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1answer
35 views

Norm of linear operator

Given two real numbers $\alpha$ and $\beta$, consider the linear operator $T:\mathbb{C}\rightarrow \mathbb{R}$ defined by $T(x+iy)=\alpha x +\beta y$. I am trying to figure out the norm of this ...
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36 views

Existence of extension from a subspace to whole space

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{span \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $Lip_0(X)$. The set Lip$_0(X)$ is the set of all real-valued ...
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1answer
48 views

Show that $T^*$ is weak$^*$-to-weak$^*$ continuous

Suppose $Y$ is a separable Banach space and $(y^*_n)_{n \in > \mathbb{N}} \subset B_{Y^*}$. Let $T : X \rightarrow Y$ be an operator. Since $Y$ is separable, we have the unit dual ball ...
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34 views

Definition of renorming of a space

The following is Proposition $4.5$ from Kalton's paper. Let $X$ and $Y$ be Banach spaces such that there exists a Lipschitz embedding $L:X \rightarrow Y$ such that $L(0)=0$ and ...
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32 views

Show that a positive homogeneous uniform continuous map is Lipschitz

Suppose $X$ and $Y$ are Banach spaces. Define a map $f:X \rightarrow Y$ which is uniform continuous and satisfies $$f(\lambda x) = \lambda f(x)$$ for $x \in X$ and $0<\lambda \in \mathbb{R}$. ...
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47 views

Definition of $\nabla f(ux_n - S(t))$ on a Banach space

Suppose $X$ is a Banach space and $f$ is a Lipschitz Gateaux differentiable function on $X$. Let $S : [0,1]^{\mathbb{N} \backslash \{ n \}} \rightarrow X$. Then we have $$f(x_n + S(t)) - f(S(t)) ...
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show that $(y^* \circ F) * \gamma(x) = \int_X {y^* \circ F(x+t)}d\gamma(t)$ is Gateaux differentiable everywhere

Suppose $X$ is a separable Banach space and $Y$ is a Banach space. If $F:X \rightarrow Y$ is a Lipschitz map and $\gamma$ is a nondegenerate Gaussian probability measure on $X$ with mean $0$, ...
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1answer
29 views

Norm of product of linear operators in normed vector space

Let $\Lambda:X\rightarrow Y$ be a linear operator, where $X$ and $Y$ are two normed vector spaces. From the definition of an operator's norm, it is straightforward that $$ \|\Lambda(x)\|=\|\Lambda ...
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32 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
32 views

Functional characterization of the boundary of a convex set

I have a closed convex set $C$ with non-empty interior in a Banach space X. I try to find a functional characterization of the boundary of $C$ in the sense that I would like to associate to $C$ a ...
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1answer
58 views

Show that a function is nowhere Gateaux differentiable.

Suppose $X$ is a Banach space. For any $x \in X$, define the set $\mathcal{F}(X) = \overline{span \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $Lip_0(X)$. The set Lip$_0(X)$ ...
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31 views

Prove the completess of the normed space in Open mapping Lemma

Here is a version of Open Mapping Lemma given in class: Let X be a Banach Space, Y a normal space and T be a bounded linear map from X to Y. Assume that for some $0 \leq \delta < 1$ and $M ...
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Extension of a linear map

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{span \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $Lip_0(X)$. The set Lip$_0(X)$ is the set of all real-valued ...
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98 views

Structure of a set whose image through continuous convex functions is an interval

What can be said about a subset $S$ of a Banach space $X$ with the property that $f(S)$ is an interval, for every convex lower semi-continuous function $f:X\to\overline{\mathbb{R}}$ that has at least ...
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45 views

If $X$ is separable, then $\mathcal{F}(X)$ is also separable, where $\mathcal{F}(X) = \overline{span \{ \delta_x : x \in X \}}$

Suppose $X$ is a Banach space. For any $x \in X$, define the set $\mathcal{F}(X) = \overline{span \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ Lip$_0(X)$. The set Lip$_0(X)$ ...
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24 views

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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2answers
114 views

$C(K)$ is reflexive if and only if $K$ is finite

Let $C(K)=$ all continuous complex valued functions on compact Hausdroff $K$. Is it true that K must be finite if $C(K)$ is reflexive? To me it seems true, but I don't know how to prove it. As $C(K)$ ...
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Separable, reflexive space that is almost isometrically universal for the class of finite dimensional normed spaces.

My question is whether there is a separable, reflexive Banach space $X$ s.t. for any finite dimensional normed space $E$ and ${\epsilon>}0$, there is a linear isomorphism $T$ from $E$ to its image ...
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What is the norm of an operator $T:c_0\to\ell^\infty$?

I have the operator $T:c_0\to\ell^\infty$ defined by $$T(x)=\left(\sum_{j\geq1}a_{ij}x_j\right)_{i=1}^\infty$$ where $\sup_{i\geq1}\sum_{j\geq1}|a_{ij}|<\infty$ such that $T(x)\in\ell^\infty$. I ...
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1answer
47 views

The coordinate-wise limit of a sequence in the unit ball of $\ell^p$ is also in the unit ball

Let $1\le p<\infty$, and let $U$ be the open unit ball in $\ell^p$. Let $\{x_k\}\subseteq\overline U$ be a sequence such that for every $n$ the limit $x(n):=\lim_{k\to\infty}x_k(n)$ exists. Can I ...
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21 views

Norm-closed subalgebras of $B(L^p(X,\mu))$

If $A$ is a Banach algebra that is isometrically isomorphic to a norm-closed subalgebra of $B(L^p(X,\mu))$ for some $p\in[1,\infty)$ and some measure space $(X,\mu)$, and is also isometrically ...
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1answer
26 views

Show that being complemented in its bidual is an isomorphic invariant

Suppose $X$ and $Y$ are Banach spaces. If $X$ are $Y$ are isomorphic and $X$ is complemented in $X^{**}$, then $Y$ is complemented in $Y^{**}$. I am self-studying the problem $1.2$ from here. $A$ ...
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Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is.

Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is. I really didn't get it. Of course both spaces are normed spaces but ...
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133 views

Definition of a norming set

In this paper, page $131$, in the proof of Proposition $4.1$, there is this sentence: We first note that if $V$ is a separable Banach space, the subset of Lip$(V)$ consisting of all weakly ...
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24 views

Compact operator in terms of exact sequences

I know a pretty equivalent definition for Fredholm operators in terms of exact sequences. Here is it: We called operator $S : E \to F$ between Banach spaces $E$ and $F$ as Fredholm iff exist exact ...
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106 views

$C[0,1]$ doesn't contain a complemented subspace isomorphic to $l^1$

I want to prove the following fact: $C[0,1]$ doesn't contain a complemented subspace which is isomorphic to $l^1$ Here is the definition of complemented subspaces. All I can do with ...
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1answer
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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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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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Stochastic integral and weak integral

Can the stochastic (Skorokhod) integral be seen as a special case of the weak of Pettis integral with the Banach space which win integrate into chosen appropriately?
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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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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 ...
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51 views

Unit ball in a separable Banach Space

Suppose that $X$ is a separable Banach space. Then there exists a sequence $\{ x_n: n \in \mathbb{N} \}$ such that the sequence is dense in a unit ball $B_X$. Question: Whenever we talk about ...
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64 views

Application of Banach Fixed Point Theorem to sequences

I have the following question Show that there is unique real bounded sequence $(a_n: n \in \mathbb{N})$ such that $$a_n = \frac{n+1}{n}+\sum^\infty_{m=1}\frac{\sqrt{3a^2_{m+n}+1}}{4m^2}$$ for all $n ...
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22 views

If the bounded operators $X\to Y$ form a Banach space in the operator norm, is $Y$ necessarily Banach? [duplicate]

I have seen that if $Y$ is Banach, the set $B(X,Y)$ of bounded linear operators from $X$ to $Y$ is Banach in the operator norm. I was now wondering about the converse. Is it true? More precisely: ...
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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 ...
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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: ...
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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 ...
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1answer
61 views

Existence of the solution in Banach space.

I have some difficulties with solving that problem: Let $X$ be a linear space and $X= \left \{u \in C([0,\alpha]):\|u\|=\sup \frac{|u(x)|}{x}\right \}$. We have a differential equation: ...
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2answers
47 views

Duality in finite-dimensional normed spaces

Suppose we endow $\mathbb{R}^n$ with a norm $\|\cdot\|$; call such a normed space $X$. Then, as a vector space, the dual space $X^*$ is also $\mathbb{R}^n$. Let $x\in X$ and $f\in X^*$. Consider the ...
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1answer
20 views

Projection maps non-zero element to non-zero element?

The following screenshot is taken from the book 'Topics in Banach Space Theory'. In the third line of the proof above, I don't understand why $0 \neq y \in Y \Rightarrow S_N(y)\neq0$. Can't we ...
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1answer
64 views

Unbounded Operator of Finite Rank

Can one find an example of an unbounded operator on a Banach space whose rank is finite? In particular, let $X, Y$ be two Banach spaces and and $T:X\to Y$ be a linear map between them such that ...