1,343 reputation
414
bio website mathtm.blogspot.com
location University of California Los Angeles, CA
age
visits member for 2 years, 3 months
seen Aug 18 at 8:50

I am a recent graduate in applied mathematics at UCLA and currently trying to break into the quantitative investment/trading industry while also continuing to pursue advanced graduate-level mathematics as both a hobby and career necessity.


Jul
2
awarded  Curious
Jul
2
awarded  Inquisitive
Jun
22
comment Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
For the example sequence of continuous functions of $L1$ norm $1$ we have $|\phi(f_{n})|\to\infty$ as $n\to\infty$ by construction of the Dirac functions outlined above. Do you see this at least? Therefore, since $||\phi||\geq|\phi(f_{n})|$ for all $n$, the claim follows.
Jun
22
comment Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
I'm not sure why you're being asked such a question if you don't follow (b). Dirac functions to which I am referring are nonnegative bump functions that spike near the origin and approach 0 elsewhere (rapidly), but such that their total integral is always $1$. In particular, they belong to the class of functions you must consider in evaluating the $\sup$ appearing in the definiton of $||\phi||$.
Jun
22
awarded  Citizen Patrol
Jun
22
comment Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
If you followed hint you would quickly see $||\phi||=\infty$ since the example sequence of sample norms in the hint is unbounded.
Jun
22
comment Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
See my edit for (a). I made a slight oversight and you are correct about your comment.
Jun
22
revised Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
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Jun
22
revised Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
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Jun
22
answered Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
Jun
22
answered Local minimal about x=0
Jun
6
comment (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
Thanks Saz! I'll have to check this textbook out. You make a lot of references to it in your answers.
Jun
6
accepted (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
Jun
6
revised Function defined by an integral.
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Jun
6
answered Function defined by an integral.
Jun
5
revised (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
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Jun
5
revised (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
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Jun
5
revised (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
added 194 characters in body
Jun
5
asked (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
May
29
comment If $f$ and $g$ are integrable functions on $I=[a,b]$ and if $h(x):=\inf(f(x), g(x))$ for all $x \in I$, prove that $h$ is integrable at $I$.
$P=\{x_{0},x_{1},\ldots,x_{n}\}$. It could be that, say, $P_{1}=\{x_{0},x_{3},x_{5},x_{11},\ldots,x_{n-1}\}$ and $P_{2}=\{x_{1},x_{2},x_{4},\ldots,x_{n}\}$. Then $P_{1}$ and $P_{2}$ are partitions of $I$ and they are sub-partitions of $P$ in the sense that $P=P_{1}\cup P_{2}$. The index sets in this example are $I_{1}=\{0,3,5,11,\ldots,n-1\}$ and $I_{2}=\{1,2,4,\ldots,n\}.$ The manner in which the sub-partitions are obtained depend on $f$ and $g$ in accordance to the construction in the solution above. Perhaps the notation of $I_{j}$ is confusing-it has nothing to do with the interval $I$