Homework Statement
Let E be a dense linear subspace of a normed vector space X, and let Y be a
Banach space. Suppose T0 \in £(E, Y) is a bounded linear operator from E to Y.
Show that T0 can be extended to T\in £(E, Y) (by continuity) without increasing its norm.
The Attempt at a Solution...
Sorry, but I disagree.
1. If you look at the proof of the squeeze theorem, I changed the \int_a^b to \int_0^{\frac{1}{4\varepsilon}}. Since p and q are integrable over any [a,b] (a, b \in\mathbb{R}), p and q are also integrable over the interval I picked. \varepsilon here only means that...
Thank you! But I can not see what is wrong with my proof. Yes, \epsilon is used to split the interval and to construct the condition in the squeeze theorem. However, these two uses are consistent, because for any 0<\epsilon<1 we can always split the interval like this. Anyway, I am not...
Hello Micromass. Here is the integral version of the squeeze theorem. Basically, that theorem says if you can show the function is between two integrable functions and the integral of the difference of those two functions is smaller than any positive number, then the function in between is also...
Homework Statement
Homework Equations
Prove that f is integrable over [0,1]
The Attempt at a Solution
I have seen some quite complicated solution on the web. For example...
Hello, brydustin. Thank you for your detailed reply. I have finished the exercise by using the following theorem. If g is continuous and f is different from g at a finite number of points then f is integrable and the integral is the same as ∫g. Of course, here we are talking about the...
Homework Statement
Homework Equations
The Attempt at a Solution
It is very easy to see intuitively that \int_0^1 f=1, because the endpoint x=1 has no impact on the integral. If we see the integral as the area under the curve, then the area under f(1) is zero. However, we are required to prove...
Hi Micromass. Thank you for your help with https://www.physicsforums.com/showthread.php?t=451462
I now know how to solve the first question, but I still do not know know to link them with e^{mn+c}. I tried the mean value theorem of integral without success. Could you please give me another...