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 […]

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

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 […]

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 […]

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?

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 […]

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. […]

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 […]

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.

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 […]

Intereting Posts

Algebraic numbers that cannot be expressed using integers and elementary functions
Solve equation $4^x-3\cdot6^x+2\cdot9^x=0$
Let V be a vector space. If every subspace of V is T-invariant, prove that there exist a scalar multiple c such that T=c1v
Are some numbers more likely to count conjugacy classes than others?
Eigenvalues of an operator
Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism?
Isomorphism from $\mathbb{C}$ to $\prod_{i=1}^h M_{n_i}(\mathbb{C})$.
Irrationals: A Group?
Gosper's unusual formula connecting $e$ and $\pi$
Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?
In any Pythagorean triplet at least one of them is divisible by $2$, $3$ and $5$.
Intuitive reasoning why are quintics unsolvable
Hockey-Stick Theorem for Multinomial Coefficients
Comparing weak and weak operator topology
How to find $\lim\limits_{x\to0}\frac{e^x-1-x}{x^2}$ without using l'Hopital's rule nor any series expansion?