I need help to prove $\lVert f\rVert := (f(a)^4+f(b)^4)^{1/4}$ is a norm on X where is the set of monotone functions on the interval [a,b]. I'm stuck with the triangle inequality and they gave us a hint to use $h(t) = (1+t^4)^{1/4}-1-t$ for t>0 and not use norm 4 on R2. any help?
#proof is a norm
grave yarrow
crimson badgeBOT
st123
tiny compassBOT
I need help to prove $\lVert f\rVert := (f(a)^4+f(b)^4)^{1/4}$ is a norm on X wh...
- Do not ping the Moderators, unless someone is breaking the rules.
- Do not ping the Helper Moderators, unless there is a conflict between helpers.
- Do not ping other members randomly for help.
- Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
- Wait patiently for a helper to come along.
- If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:
+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:
vale zealot
$$1+t^4 \leqslant (1+t)^4$$
crimson badgeBOT
aL
vale zealot
the function h is nonpositive for t > 0
your goal is to show
$$|f+g| = \left ( (f+g)(a)^4 + (f+g)(b)^4 \right)^{1/4} \leqslant \left ( f(a)^4 + f(b)^4 \right)^{1/4} + \left ( g(a)^4 + g(b)^4 \right)^{1/4}$$
crimson badgeBOT
aL
grave yarrow
actually there was a mistake and they changed the question -
X is the sub-space of C([a,b]) spanned by $cos(\frac{\pi}{b-a}\cdot (x-a))$ and $\frac{3}{b-a}(x-a)-1$ and then the rest is ok
crimson badgeBOT
st123
balmy walrusBOT
@grave yarrow
vale zealot
@grave yarrowclose the topic