Taylor Martin
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 Jun22 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 $||\phi||$. Jun22 awarded Citizen Patrol Jun22 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. Jun22 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. Jun22 revised Computing the norm of operator when space is equipped with sup norm and $L^1$ norm added 242 characters in body Jun22 revised Computing the norm of operator when space is equipped with sup norm and $L^1$ norm deleted 73 characters in body; added 177 characters in body Jun22 answered Computing the norm of operator when space is equipped with sup norm and $L^1$ norm Jun22 answered Local minimal about x=0 Jun6 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. Jun6 accepted (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$ Jun6 revised Function defined by an integral. added 293 characters in body Jun6 answered Function defined by an integral. Jun5 revised (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$ added 3 characters in body Jun5 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 Jun5 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 Jun5 asked (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$ May29 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 $I$ May29 revised 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 $I$. added 810 characters in body May29 revised 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 $I$. added 810 characters in body May29 revised 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 $I$. added 810 characters in body