In this paper, we have gone over briefly the three areas which David Hilbert was famous for. His teaching ability, work on Hilbert Spaces, and his 23 mathematical problems. He was without a doubt an incredible mathematician and changed the way we see mathematics forever. His problems were studied by countless century mathematicians.

Hilbert Spaces are a generalization of Euclidian space, from one and two dimensions to infinite dimensions. There are four main operations that are necessary to determine a Hilbert Space, it must be a linear vector space, have a valid inner product, be separable and be complete. Hilbert Spaces lead the way for quantum mechanics, I would have liked to go into more detail on where Hilbert Spaces are used, and how it changed quantum mechanics if further space and time allowed it. I also would have liked to include how I could have extended the introduction of Hilbert Spaces to show how it could be orthogonal, and how it leads to helping with the Riesz representation theorem.

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Hilbert’s 23 problems are considered one of if not the most challenging and important problem lists ever put forward. The impact it had on the entire world of mathematics is hard to comprehend. With there still being 3 unsolved problems I would have liked to have delved a little deeper into why they are still unsolved, however, the mathematics is very complicated and hard to explain. Smale’s problems and the millennium prize problems are also very influential problem lists. They were proposed in the 21st century and included some of Hilbert’s original problems.

I wanted to compare the differences in problems mathematicians faced in the 19th century and the 21st century because I found it interesting how even over two centuries Hilbert’s problems are still unsolved. I wanted to see if the ability and difficulty of solving his equations got easier as technology, and communication developed throughout the century. I found that in my opinion, mathematics has changed massively due to these developments. Many of the solutions to Hilbert’s problems were achieved with the help of computers or collaborations from other mathematicians from the other side of the world.