Whilst writing out revision notes, something struck me!

let be a **finite dimensional** real inner product space.

let be a subspace of V.

#### Definition

The **annihilator** of , denoted where is the dual space of V.

#### Definition

The **perpendicular space** of , denoted

It might already be apparent that there is some connection between these two – both involve things going to zero, whenever something is done to everything in .

To make this more explicit, first we need a theorem.

#### Theorem: **(weaker version of) Riesz Representation Theorem**

is an isomorphism of vector spaces. Restated, we can associate each linear map uniquely with a vector such that .

#### Proof:

##### Linearity:

let

and hence we see this is a linear map.

##### Injectivity:

Suppose

then

so specifically

thus by positive definiteness, hence is injective.

##### Surjectivity:

Since we have that this is an injective linear map between spaces of the same dimension, this follows by Rank Nullity. Alternatively:

let

take a basis

then define

and we can check that this works.

Now to use this theorem to show the connection.

but for some .

thus

So we have the result we wanted:

## Applications

**This means all of those oh-so-fun identities one derives for annihilator subspace structure can be passed over to perpendicular subspace structure!**

#### Example:

Given the identity

we can deduce

linearity of gives us

so by injectivity of we have

i.e. compare

to