• Support PF! Buy your school textbooks, materials and every day products Here!

Hilbert Spaces

  • Thread starter Kindayr
  • Start date
  • #1
161
0

Homework Statement



Let [itex]H[/itex] be a Hilbert space. Prove [itex]\Vert x \Vert = \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert}[/itex]


The Attempt at a Solution


First suppose [itex]x = 0[/itex]. Then we have [itex]\sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert (0,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert 0 \vert}{\Vert y \Vert} = 0 = \Vert 0 \Vert[/itex].

Now suppose [itex]x \neq 0[/itex]. Then [itex]\Vert x \Vert = \sqrt{(x,x)} = \frac{\sqrt{(x,x)} \cdot \sqrt{(x,x)}}{\sqrt{(x,x)}} = \frac{\vert (x,x)\vert}{\Vert x \Vert} \leq \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert}[/itex].

Now I just can't do the reverse inequality. Any help is much appreciated.
 

Answers and Replies

  • #2
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,394
179
Cauchy-Schwarz?
 
  • #3
161
0
Cauchy-Schwarz?
omg how do i call myself a math major.

thank you.
 

Related Threads for: Hilbert Spaces

  • Last Post
Replies
12
Views
3K
  • Last Post
Replies
3
Views
828
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
19
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
12
Views
1K
  • Last Post
2
Replies
42
Views
5K
Top