Articles of banach spaces

Absolutely convergent sums in Banach spaces

Let’s say a sum of elements in a Banach space is absolutely convergent if even the sum of the norms converges, i.e. $\sum_{i=1}^\infty ||x_i|| \le \infty$. This condition implies that the sum of the $x_i$ can be reordered, and in fact that it can be represented in this way that has no dependence on the […]

Normed space where all absolutely convergent series converge is Banach

This question already has an answer here: If every absolutely convergent series is convergent then $X$ is Banach 2 answers

Continued matrices-valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. I’d like to construct a continuous function $K:\mathbb{R}^k\rightarrow {\cal M}_{k\times (k-d)}(\mathbb{R})$ such that $K(x)$ is full rank […]

Orthogonal in inner product space

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal then$||x+αy||^2=||x||^2+|α|^2||y||^2 \ge ||x||^2 $ so $||x+αy|| \ge ||x||$ . But the second direction if $||x+αy|| \ge ||x||$ for any scalar […]

Frechet differentiable implies reflexive?

Note: The question has been cross-posted (and answered) on MathOverflow here. Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive?

$C ( \times \to \mathbb R)$ dense in $C ( \rightarrow L^{2} ( \to \mathbb R))$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense in $L^{2} ([0,1] \to \mathbb R)$ (the usual Lebesgue space). Now consider the space of continuous functions on $[1,2]$ taking […]

Examples of double dual spaces

I am looking for examples on double dual spaces. As I know $\ell_p $ is the double dual of $\ell_p$ for $1<p,q<\infty$, $\mathcal L_p $ is the double dual for $\mathcal L_p$ for $1<p,q<\infty$ and $\ell_\infty$ is the double dual of $c_0$, where $c_0$ is the space of sequences of numbers that converge to 0. […]

Banach-Space-Valued Analytic Functions

This is Chapter VII, $\S$3, exercise 4, from Conway’s book: A Course in Functional Analysis: Let $X$ be a Banach space and $G\subset \mathbb{C}$ an open subset. We say that $f: G \to X$ is analytic if the limit $$\lim_{h \to 0} \frac{ f(z+h)-f(z)}{h}$$ exist in $X$ for all $z \in G$. Prove that if […]

Isometry from Banach Space to a Normed linear space maps

Let $X$ be a Banach Space and $Y$ be a normed linear space. Show that if $T$ is an isometry then $T(X)$ is closed in $Y$. Let me have some idea to solve this. Thank you for your help.

Prove that a linear operator $T:E \rightarrow E'$ such that $\langle Tx,y \rangle=\langle Ty,x\rangle$ is bounded

Let E be a Banach space and $T:E\to E’$ a linear operator such that $\langle Tx,y\rangle=\langle Ty,x\rangle$ for all $x,y\in E$. Here $E’$ is the dual space of $E$. I have to prove that $T$ is a bounded operator. I tried to use the closed graph theorem, but I can’t prove that the graph of […]