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Banach space for Frobenius norm

Mathematics Asked by Dong Le on January 11, 2021

I want to prove that $(mathcal{M}_{mtimes n}(mathbf{R}), ||.||_F)$ is a Banach Space

where
$A in mathcal{M}_{mtimes n}(mathbf{R})$, the Frobenius norm:

$$
||A||_F = sqrt{sum_{i = 1}^{m} sum_{j=1}^{n} a_{ij}^{2} }
$$

I attempted to redefine the matrix $A$ into linear mapping $T: mathbf{R}^{n} to mathbf{R}^{m}$, where $T(x) = Ax$ for $x in mathbf{R}^{n}$.

I am stuck on showing a Cauchy Sequence $(T_n) subset mathbf{R}^{m}$ is converging to $T$ with respect to the norm $ ||. ||_F$ . How should I go from this approach or any suggestion differently?

One Answer

So, should I define the Cauchy sequence of matrices in this way:

$$ A^{(n)} = begin{bmatrix} x_{11}^{(n)} & dots & x_{1n}^{(n)} \ vdots & vdots \ x_{m1}^{(n)} & dots & x_{mn}^{(n)} end{bmatrix} $$

where each entries in these matrices are Cauchy Sequence. By Classical Convergence Cauchy criterion in $mathbf{R}$, it converges. So the Cauchy sequence of matrices converge to $A$ with respect to $||.||_F$.

Answered by Dong Le on January 11, 2021

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