# Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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### Prove norm does not come from inner product.

I know I have to show it does not satisfy the parallelogram law but I don't know how to apply it.
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### Prove norm doesn't come from inner product.

Please help me prove this. I'm not sure how to apply the parallelogram law to the norm.
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### Test for normability of a metric on a Banach space

If d is a metric on a (finite dimensional) Banach Space and there exist norms $\|-\|_1$ and $\|-\|_2$ and constants $C_1,C_2 \in [1,\infty)$ satisfying: C_1\|x-y\|_1 \leq d(x,y) \leq ...
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### Equivalence of norms: $\|x\|$ and $\|x\|_1=\|x\|+\lvert\,f(x)\rvert$

Let $f: X\to \mathbb{R}$ be a linear functional and $\|\cdot\|_1$ is defined as follow $\|x\|_1=\|x\|+|f(x)|$. Prove or disprove $\|\cdot\|_1$ is equivalent to $\|\cdot\|$ iff $f$ is continuous. I've ...
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### Difficulty in understanding a step in a definition in the book Walter Rudin

In the book Principles of mathematical analysis by Walter Rudin,He writes: "For $A\in L(\Bbb R^n,\Bbb R^m)$, define the norm $||A||$ of $A$ to be sup of all numbers $|Ax|$, where $x$ ranges over all ...
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### Subset of the sequence space that's closed and bounded but not compact

Consider the sequences space $l^1 = \{a = (a_n)_{n \in \mathbb{N}_0} \subset \mathbb{C}, \sum_{n = 0}^\infty|a_n|< \infty\}$ with the norm $||a||_1 = \sum_{n = 0}^\infty|a_n|$. I want to show that ...
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Show: If $A$ is a symmetric matrix and $\|A\|_F\leq1$, then prove that $I-A$ is a positive semidefinite matrix.
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Assume we have a postive definite real matrix $P$ and we define an inner product on a finite dimensional hilbert space $\langle x, y \rangle = x^\top P y$ and clearly the induced norm is $|| x || = \... 1answer 46 views ### conditions for norm of linear bounded operator to satisfy$\lvert T_x (y) \rvert = \lVert T_x \rVert$. Let$x = (x_n)_{n \in \mathbb N} \in l^\infty$and let$T_x : l^1 \rightarrow \mathbb F$be defined by$T_x (y) = \sum_{n=1}^\infty x_ny_n$. What condition on$x$is needed so that there exists$y \in ...
I'm trying to prove that $||AB||_F\leq||A||_2||B||_F$. As far as I know it isn't a hard problem but I was stuck. Any ideas?