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### Boundedness of a matrix operator in a norm

As explained in the comments, you are mixing up two different concepts: boundedness of a matrix and uniform boundedness of a family of matrices. To illustrate this via the example \begin{align*} A:(0,\...
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1 vote
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### Norm preservation properties of a unitary matrix

Let $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ be either the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. Definition (Unitary matrix). A unitary matrix is a square ...

### Metric on $\mathbb R^{m \times n}$ based on the Frobenius norm

$\def\R{\mathbb{R}}\def\T{^{\mathrm{T}}}\def\ge{\geqslant}\def\le{\leqslant}\def\peq{\mathrel{\phantom{=}}}\def\norm#1{\left\lVert#1\right\rVert}\def\paren#1{\left(#1\right)}$Proposition 1 (Ptolemy): ...
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### For a given $k \in \mathbb{N}$, define the mapping $A: X \to X$ by the rule $(Af)(x) = x^k f(x).$

To show that $A$ is linear means to show that $$A(\alpha f+\beta g)=\alpha Af+\beta Ag$$ or, equivalently, that $$x^k(\alpha f(x)+\beta g(x))=\alpha x^kf(x)+\beta x^kg(x).$$ This identity clearly ...
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### I want to prove that $T$ belongs to $B(X)$, meaning it is a bounded linear operator.

I think that $f \in X^*$ should be a bounded linear operator as well. I'm assuming it is a operator of this kind: $f:X\to\mathbb{K}$ where $\mathbb{K}$ is a field equal to $\mathbb{R}$ or $\mathbb{C}$...

### Metric on $\mathbb R^{m \times n}$ based on the Frobenius norm

The following is a partial answer as it only fully addresses your second question. As for the first question: while numerics suggest that your map $d$ is indeed a metric, I cannot even prove the ...
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### Nature of the Euclidean Norm

This is subtle. In mathematics we have the freedom to define distances however we want depending on the application, and for example sometimes rather than the Euclidean norm we use what are called the ...
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### Nature of the Euclidean Norm

The definition is partly "an greed-upon convention", but at the same tame there is a reason to define it the way it is. There is mathematical notion called "metric" or "...
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