# Tagged Questions

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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### Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? Context: ...
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### Closed map $T:X \to Y$ has closed graph?

Let $T:X\to Y$ be a linear operator between two normed vector spaces. My question is: If $T$ is a closed map (sends closed sets to closed), then is the graph of $T$ a closed set of $X \times Y$? ...
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### What is this operator topology?

Let $X$ be a separable Banach space with (norm $1$) Schauder-Basis $\{e_n\}_{n\in\mathbb N}$. Denote for $x\in X$ with $|\cdot|_x$ the seminorm on $\mathcal L(X)$ given by $|A|_x = \|A x\|$. Consider: ...
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### Linear Operators: Continuous $\Rightarrow$ Bounded

Let $T:V\rightarrow V'$ be a continuous linear operator between two normed vector spaces $V,V'$. Show that it is bounded. Continuity is defined as $\lim_{n}\|x_n-x\|=0\Rightarrow \lim_n\|T(x_n-x)\|=0$....
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### Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1})$ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
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### Neumann series in an incomplete normed algebra

Let $\mathcal{A} \equiv (A, \|\cdot\|_A)$ be a unital (associative) normed algebra over the real or complex field, and assume that $\mathcal{A}$ is not complete. Provided $\mathcal{B}_\mathcal{A}$ is ...
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### How did Von Neumann think of the formula for scalar product?

My question is on the idea behind Von Neumann's formula for the scalar product induced from a norm that satisfies the parallelogram law. $\langle x,y\rangle=\frac 14(\|x+y\|^2-\|x-y\|^2)$. Was it by ...
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### Operator norm of matrix of scalars regarded as matrix with entries in the unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+=\{(a,z):a\in A,z\in\mathbb{C}\}$ with product $(a,z)(b,w)=(ab+zb+wa,za)$ and norm $||(a,z)||=||a||+|z|$. Equip $M_n(A^+)$ with the operator norm by ...
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 ...