# 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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### Sum of open and closed sets

Let $A,B$ subsets of a normed space $(X,\|\cdot\|)$ and $A+B=\{a+b\mid a\in A,\, b\in B\}$ I need help with the next proofs, I can't figure how to begin the proofs: (a) If $A,B$ open then $A+B$ open ...
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### Bounded vectors give bounded scalars, finite-dimensional vector space.

Assume that we are in a finite dimensional vector space, with basis $\{v_1,v_2,\ldots,v_n\}$. Assume also that we have a sequnce of bounded vectors, $\{x_i\}$, that is $\|x_i\|<M$ for some real ...
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### Define $g :\ell_2 \to \mathbb R$ by $g(x)= \sum_{n=1}^{\infty} \frac{x_n}n$. Is $g$ continuous?

Define $g :\ell_2 \to \mathbb R$ by $$g(x)= \sum_{n=1}^{\infty} \frac{x_n}n$$ Is $g$ continuous? I need to solve this but I could not see how to tackle it? any hints or suggestion?
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### Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
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### Norms on unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+$ be its unitization. Then $||(a,z)||_1=||a||+|z|$ is a Banach algebra norm on $A^+$. Can we also make $A^+$ a Banach algebra by giving it the norm ...
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### Show that $\exists$ an inner product in $X$ such that $<x,x> =||x||^2$ for all $x \in X$

Problem: Let $X$ be normed space. If on every two dimensional subspace $Y$ of $X$, there is an inner product $<,>_Y$ such that $<y,y>_Y=||y||^2$ for all $y\in Y$. Then there is an inner ...
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### Confused by peculiar norm

Let $X$ be an infinite subset of $[0,1]$. In an exercise I am considering the norm on $P([0,1])$ (polynomials on unit interval) defined by: $$||p||_X=\sup_X |p|$$ My question is, how do I make sense ...
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### Uncountably many norms such that no two are Lipschitz equivalent

I am struggling with the following question: Is it possible to find uncountably many norms on $C[0,1]$ such that no two are Lipschitz equivalent? I had thought about trying to define norms for each ...
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### Proving that given this property then the norm is induced by a inner product

Let $(X,||\cdot||)$ be a normed space such that, for $x,y\in X$ $$||x + y||^2 +||x - y||^2 = 2||x||^2 + 2||y||^2$$. Then I want to check that $||\cdot||$ is induced by an inner product, so what I ...
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### Why aren't the rationals a compact subset of $\mathbb{R}$?

We define a compact subset of some normed vector space $V$ to be any subset $S$ where every sequence $\{\mathbf{x}_{n}\}$ in $S$ has a subsequence which converges to some $\mathbf{x}$ in $S$. Then ...
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### dual pairs in $\mathbb R^n$ with non-standard norms $l_1$ and $l_{\infty}$

Suppose $X=Y=\mathbb R^n$. Usually, to apply a separating hyperplane theorem in $X$ (the primal space), we associate both $X$ and $Y$ with the standard Euclidean norm. My question is that could I ...
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I'm interested in the properties of countable basis functions that span functions living in $\Bbb R^3$. Can I represent a $L^2$ normalizable function that has a point divergence, (for example, $\... 1answer 41 views ### Is it possible to strengthen this inequality? Let$T:X\to Y$be a linear operator from a normed space$X$into a normed space$Y$. Suppose that$T$has the property that for a fixed$y\in Y$and any$\alpha>1$, there exists an$x_{\alpha}\in ...
Are the following true? $\operatorname{scalar} \circ \operatorname{function} = \operatorname{scalar} \times \operatorname{function}$ $\operatorname{function} \circ \operatorname{scalar} =$ the ...
### 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 $n$.