# Functional Analysis question

• May 1st 2010, 02:38 AM
blimp
Functional Analysis question
I can't work this out

show \$\displaystyle c_0\$ (with the usual sup norm) is not a Hilbert Space.

my main problem stems from using the \$\displaystyle c_0\$ space which I don't fully grasp, I know the method for this type of problem so I'm really looking for a suggestion as a what to let x and y equal and what their norms should look like
• May 1st 2010, 07:28 AM
Opalg
Quote:

Originally Posted by blimp
I can't work this out

show \$\displaystyle c_0\$ (with the usual sup norm) is not a Hilbert Space.

my main problem stems from using the \$\displaystyle c_0\$ space which I don't fully grasp, I know the method for this type of problem so I'm really looking for a suggestion as a what to let x and y equal and what their norms should look like

The way to show results like this is to use the parallelogram identity. You can choose almost any two elements of the space to see that they do not satisfy the identity. The easiest choice would be to take for example x to be the sequence in \$\displaystyle c_0\$ having a 1 for its first coordinate and 0 for every other coordinate; and take y to be the sequence having a 1 for its second coordinate and 0 for every other coordinate. Then x, y, x+y and x–y all have \$\displaystyle c_0\$-norm 1