Orthonormal basis in a Hilbert space

Question

My understanding of an orthonormal basis is in Cartesian coordinates. Essentially, every coordinate (x,y,z) has a unit vector, then any other vector in this space can be defined by the unit vectors (x,y,z). To my knowledge, the orthonormal basis vectors need to be orthogonal to each other. My question comes to when we consider orthonormal basis vectors in a Hilbert space. It seems in a Hilbert space you can have any number of dimensions rather than just three in the Cartesian. If we have infinite dimensions, how can the basis vectors be orthogonal? Also, in a 3d sphere, we can use spherical coordinates to describe a vector, would this be an example of a 3d vector in a Hilbert space? Then anything past 3d is impossible to imagine?

Answer 1

The basis vectors need not be orthonormal. They just need to be linearly independent. That just means you can't write one of them as a scalar multiple of the other. Visually, linearly independent vectors aren't colinear. So you can have two vectors, (0,2) and (1,3) in a plane for example, as your basis vectors. A two dimensional vector can be thought of as a list of two numbers like $(0,1)$. A three dimensional vector is a list of 3 numbers. Generalizations to higher dimensions can be easily made in this way. However, the concept of a vector is probably a lot deeper than a list of numbers, and definitely a lot more rigorous than that. The dimension of the space doesn't have a mathematical restriction on it to be 3 (nor a physical restriction for that matter). Orthogonality is defined by the inner product of two vectors. The vectors themselves can be two dimensional, three dimensional, or n-dimensional. For example, the two vectors $A=(1,1,1,1)$ and $B=(1,1,-1,-1)$ in 4-dimensional Euclidean space are orthogonal, if you use the Euclidean inner product, giving $A.B=(1)(1)+(1)(1)+(1)(-1)+(1)(-1)=0$. It's the same for a Hilbert space. It's just another vector space that can have any given number as its dimension. That's because a vector isn't just an arrow in our three dimensional space. A vector is something much more general than that. A basic introduction to linear algebra would explain those concepts as well as remove the restrictions you're imposing on things like dimensionality and orthogonality. But basically yes, it's impossible to visualize or imagine but not impossible to write down formally using mathematics.