# 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 weakly continuous implies strong continuous? [duplicate]

Let $(X, \|\cdot\|_X), (Y, \|\cdot\|_Y)$ normed spaces, operator $A: X \longrightarrow Y$ weakly continuous. Is $A$ continuous with respect to the strong topology? Thanks in advance for your ideas.
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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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### To show $T$ bounded

Let $X,Y$ be normed linear spaces and let $T:X\to Y$ be a linear map such that for every absolutely convergent series $\sum\limits_{n=1}^{\infty}x_n$, the series $\sum\limits_{n=1}^{\infty}Tx_n$ ...
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### Norm of pointwise product of Lp functions

Does the following inequality hold in $L_p$ spaces? $\|fg\|_p\leq\|f\|_p\|g\|_p$ How would I go about proving this? Do I need to apply Cauchy Schwarz?
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### How to proof that a finite-dimensional linear subspace is a closed set

Given a linear space V, a field F, a norm $||.||$ on V and a Base B. How do i proof that the sub-space span{$b_1,b_2,...,b_n$} where $b_i \in B$ is a closed set under the topology that is created ...
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### Show that $Z$ is the null space of a suitable linear functional $f$ on $X$

If $Z$ is an $n-1$ dimensional subspace of an $n$ dimensional vector space $V$ . Show that $Z$ is the null space of a suitable linear functional $f$ on $X$ which is uniquely determined upto a scalar ...
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### Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, s.th. $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. ...
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### Convex and dense subset in infinite-dimensional normed vector space

Let $X$ be an infinite-dimensional normed vector space. How to construct proper convex subset $A$ of $X$, s. th. $A$ is dense in $X$ ? If $X$ is an unitary vector space then it is obvious, but how to ...
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### Showing the the unit sphere is closed using sequences

Let $(X,\|\cdot\|)$ be a normed space. Prove that every sequence in $S_X=\{x\in X\mid \|x\|=1\}$ converges in $S_X$. My attempt. Let $(x_n)\in S_X$. Then, $\|x_n\|=1$ for all $n$. Now assume that ...
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### Is an unbounded function bounded on a bounded non-compact interval?

I'm a little confused about functions in the set of bounded continuous functions. For example, if we take the interval (0,1] and the function $f(x) =$\begin{cases} 0 & x \in (0,\frac{1}{2}]\\...
I know that Bellman operator, defined as $T(f(x))=\sup_{y\in\Gamma(x)} {\phi(x,y)+\beta f(y)}$, is a contraction provided that $\beta\in(0,1)$ and $\phi$ is a bounded function on $Gr\Gamma$. In ...