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Jul
26
comment Are they true these generalizations from matrices to operators about functional calculus?
My pleasure. $| $
Jul
26
comment Are they true these generalizations from matrices to operators about functional calculus?
It is probably overkill, one can do it directly using the definition of the operator norm.
Jul
26
comment Are they true these generalizations from matrices to operators about functional calculus?
Exactly. ${\ \ }$
Jul
26
revised Trace: Independence
edited body
Jul
26
comment Are they true these generalizations from matrices to operators about functional calculus?
Writing limits is never a good practice. To prove limit equalities, you deal with inequalities. The key piece of information here is that, for a direct sum, $\|A\oplus B\|=\max\{\|A\|,\|B\|\}$. The block-diagonal matrix is a direct sum.
Jul
26
answered Trace: Independence
Jul
26
comment How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
Yes.${ \ \ \ \ }$
Jul
26
comment How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
With the parametrization in the answer, $dV=r\,drdtdz$.
Jul
26
comment Are they true these generalizations from matrices to operators about functional calculus?
Yes to your first question. As for the second one, you can approximate your continuous function uniformly by polynomials; as long as your measure is finite (which it is, in this case), the limit of the integrals is the integral of the limit.
Jul
26
comment How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
Yes. Please see the edit.
Jul
26
revised How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
added 188 characters in body
Jul
26
comment How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
In that case you don't want to parametrize the surface; you want to parametrize the volume inside.
Jul
26
comment How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
When you calculate a flux there is no "cartesian" or "polar". You integrate over a parametrization of your surface.
Jul
26
comment E or natural log problem, solve the equation
Yes. You solve the quadratic and you'll obtain the possible values for $e^x$, and then you solve for $x$.
Jul
26
answered Are they true these generalizations from matrices to operators about functional calculus?
Jul
26
comment How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
My pleasure. ${ }$
Jul
26
answered E or natural log problem, solve the equation
Jul
26
answered How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
Jul
26
revised Proof that |x-a| is continuous at x=a (epsilon delta), and nondifferentiable at x=a.
added 80 characters in body
Jul
26
revised Proof that |x-a| is continuous at x=a (epsilon delta), and nondifferentiable at x=a.
edited tags