Statement :- We assume the orthagonal sequence
in Hilbert space, now
, the Fourier coefficients are given by:
![{\quad \qquad \longrightarrow \sf a_{i}=(v,{\phi}_{i})}](https://tex.z-dn.net/?f=%7B%5Cquad%20%5Cqquad%20%5Clongrightarrow%20%5Csf%20a_%7Bi%7D%3D%28v%2C%7B%5Cphi%7D_%7Bi%7D%29%7D)
Then Bessel's inequality give us:
![{\boxed{\displaystyle \bf \sum_{1}^{\infty}\vert a_{i}\vert^{2}\leqslant \Vert v\Vert^{2}}}](https://tex.z-dn.net/?f=%7B%5Cboxed%7B%5Cdisplaystyle%20%5Cbf%20%5Csum_%7B1%7D%5E%7B%5Cinfty%7D%5Cvert%20a_%7Bi%7D%5Cvert%5E%7B2%7D%5Cleqslant%20%5CVert%20v%5CVert%5E%7B2%7D%7D%7D)
Proof :- We assume the following equation is true
![{\quad \qquad \longrightarrow \displaystyle \sf v_{n}=\sum_{i=1}^{n}a_{i}{\phi}_{i}}](https://tex.z-dn.net/?f=%7B%5Cquad%20%5Cqquad%20%5Clongrightarrow%20%5Cdisplaystyle%20%5Csf%20v_%7Bn%7D%3D%5Csum_%7Bi%3D1%7D%5E%7Bn%7Da_%7Bi%7D%7B%5Cphi%7D_%7Bi%7D%7D)
So that,
is projection of
onto the surface by the first
of the
. For any event, ![{\sf (v-v_{n})\perp v_{n}}](https://tex.z-dn.net/?f=%7B%5Csf%20%28v-v_%7Bn%7D%29%5Cperp%20v_%7Bn%7D%7D)
Now, by Pythagoras theorem:
![{:\implies \quad \sf \Vert v\Vert^{2}=\Vert v-v_{n}\Vert^{2}+\Vert v_{n}\Vert^{2}}](https://tex.z-dn.net/?f=%7B%3A%5Cimplies%20%5Cquad%20%5Csf%20%5CVert%20v%5CVert%5E%7B2%7D%3D%5CVert%20v-v_%7Bn%7D%5CVert%5E%7B2%7D%2B%5CVert%20v_%7Bn%7D%5CVert%5E%7B2%7D%7D)
![{:\implies \quad \displaystyle \sf ||v||^{2}=\Vert v-v_{n}\Vert^{2}+\sum_{i=1}^{n}\vert a_{i}\vert^{2}}](https://tex.z-dn.net/?f=%7B%3A%5Cimplies%20%5Cquad%20%5Cdisplaystyle%20%5Csf%20%7C%7Cv%7C%7C%5E%7B2%7D%3D%5CVert%20v-v_%7Bn%7D%5CVert%5E%7B2%7D%2B%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%5Cvert%20a_%7Bi%7D%5Cvert%5E%7B2%7D%7D)
Now, we can deduce that from the above equation that;
![{:\implies \quad \displaystyle \sf \sum_{i=1}^{n}\vert a_{i} \vert^{2}\leqslant \Vert v\Vert^{2}}](https://tex.z-dn.net/?f=%7B%3A%5Cimplies%20%5Cquad%20%5Cdisplaystyle%20%5Csf%20%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%5Cvert%20a_%7Bi%7D%20%20%5Cvert%5E%7B2%7D%5Cleqslant%20%5CVert%20v%5CVert%5E%7B2%7D%7D)
For
, we have
![{:\implies \quad \boxed{\displaystyle \bf \sum_{1}^{\infty}\vert a_{i}\vert^{2}\leqslant \Vert v\Vert^{2}}}](https://tex.z-dn.net/?f=%7B%3A%5Cimplies%20%5Cquad%20%5Cboxed%7B%5Cdisplaystyle%20%5Cbf%20%5Csum_%7B1%7D%5E%7B%5Cinfty%7D%5Cvert%20a_%7Bi%7D%5Cvert%5E%7B2%7D%5Cleqslant%20%5CVert%20v%5CVert%5E%7B2%7D%7D%7D)
Hence, Proved
Answer:
5.714 hours / day
Explanation:
<u>Calculate the hours used in that week </u>
120000 / 3000 = 120 / 3 = 40 hours a week
<u>Calculate the amount it is used in one day</u>
40 / 7 = 5.71428571 hours or 5.714 hours/day
Answer:
1. The magnetic field encircles the wire in a counterclockwise direction
Explanation:
When we have a current carrying wire perpendicular to the screen in which the current flows out of the screen then by the Maxwell's right-hand thumb rule we place the thumb of our right hand in the direction of the current and curl the remaining fingers around the wire, these curled fingers denote the direction of the magnetic field which is in the counter-clock wise direction.
Ever current carrying conductor produces a magnetic field around it.
I think its a tbh bc it seems to be the best answer out of a b c and d