Eigenfunctions orthogonal in Hilbert space

[gfd43tg]

Hello,

I am having a question regarding how eigenfunctions are orthogonal in Hilbert space, or what does that even mean (other than the inner product is zero). I mean, I know in $\mathbb {R^{3}}$, vectors are orthogonal when they are right angles to each other.

However, how can functions be "orthogonal", in the sense of being perpendicular, and does Hilbert Space have infinite dimensions?

 

[Orodruin]

I am having a question regarding how eigenfunctions are orthogonal in Hilbert space, or what does that even mean (other than the inner product is zero).

There is no other meaning to it, it simply means the inner product between the functions is zero. The geometrical interpretation for $\mathbb R^n$ is that two orthogonal vectors are at right angle to each other, but really this is also a matter of definition of orthogonality.

and does Hilbert Space have infinite dimensions?

It can have infinite dimensions, yes. It can even be non-separable. It does not have to be infinite dimensional.

 

[gfd43tg]

But how can you even take the inner product of functions? I thought this was something you did with vectors. For example, what's the inner product of $f_{1}(x) = x$ and $f_{2}(x) = x^{2}$? This doesn't mean anything to me

 

[DrClaude]

But how can you even take the inner product of functions? I thought this was something you did with vectors. For example, what's the inner product of $f_{1}(x) = x$ and $f_{2}(x) = x^{2}$? This doesn't mean anything to me

For functions, the inner product is usually defined (in physics) as
$$
\langle f, g \rangle \equiv \int_a^b f^* g \, d\tau
$$
where $a$ and $b$ are appropriate limits and the integration element $d\tau$ will depend on how the function is expressed. In 1D, $d\tau$ will usually be $dx$ or $dp$.

 

[Orodruin]

But how can you even take the inner product of functions? I thought this was something you did with vectors. For example, what's the inner product of $f_{1}(x) = x$ and $f_{2}(x) = x^{2}$? This doesn't mean anything to me

Function spaces can also be vector spaces. As long as you can add functions and multiply them by constants and still be within the function space (with all of the relevant requirements fulfilled), it is a vector space. For example, the identity vector under addition is simply the zero function, for which f(x) = 0 for all x. The inner product that DrClaude mentions fulfils all of the requirements of an inner product (sometimes it will also come with an additional weight function), which you can check by simply ticking off the axioms for an inner product. A function space which is a vector space endowed with an inner product is an inner product space. A Hilbert space is essentially an inner product space where all Cauchy sequences converge to an element in the space.