Orthogonal Complement in Inner Product Space: W2^\bot\subseteqW1^\bot

[chuy52506]

Let W₁ and W₂ be subspaces of an inner product space V with W₁$$\subseteq$$W₂. Show that (the orthogonal complement denoted by $$\bot$$) W₂^{$$^\bot$$}$$\subseteq$$W₁^{$$^\bot$$}.

 

[Landau]

You mean W1 instead of the second W2 in the latter inclusion.

Anyway, just write out the definitions. Really.

 

[Fredrik]

Just a little tip about the notation: ^\bot looks better than \bot.

 

[chuy52506]

How would you start it though?

 

[Fredrik]

You start by saying "Let $x\in W_2^\perp$ be arbitrary". Then you show that it's a member of $W_1^\perp$. You should see how to do this if you just write down the definition of "orthogonal complement".