David Hilbert was a German mathematician who was born on January the 23rd 1862 in Konigsberg, East Prussia which used to be a German State. It is now known as Kaliningrad, Russia. He was brought up with two educated parents, his father, Otto Hilbert was a city judge and his mother Maria Terese Hilbert was very interested in philosophy and astronomy. It was her love for numbers and shapes which influenced Hilbert from a young age. He was the first child of two children and the only boy in the family. Hilbert always had a strong interest in Mathematics and Languages, however, he chose mathematics and enrolled in Wilhelm Gymnasium, a very science-oriented school. In 1880, he enrolled at Königsberg University to pursue his career in mathematics, and quickly excelled while he was there. He graduated with a doctorate degree in philosophy in 1885. This was because of his hard work, dedication and natural mathematical ability. During his time at Königsberg University, he made a couple of tours to Europe to meet with some important scientists of his time, he also enrolled to Heidelberg in order to go to additional lectures and classes. This just shows how determined he was and how massive his interest in mathematics was. He also met his wife Käthe Jerosch at Konigsberg university and had a son together named Franz. Franz, unfortunately, suffered from an undiagnosed psychological illness his whole life which David Hilbert saw as a heavy burden. He saw his lack of intelligence as a massive disappointment and voiced it frequently.

In 1895, he was offered a permanent job as a professor in mathematics at the University of Göttingen. The University of Göttingen was seen as the leading school for mathematics in Germany. Many famous influential mathematicians attended the university such as Gauss who’s known mainly for his work in number theory and electromagnetism and Riemann who was mainly known for his work in number theory and differential geometry. There he became very good friends with Alonzo Church and Emmy Noether who also went on to become exceptional mathematicians. As a professor he supervised 69 students, many of them also going on to become famous mathematicians themselves. He was known for his simple, clear lectures where he commonly used examples and thought of experiments to help explain general methods. It was because of these methods of teaching that he was able to produce such high-quality mathematicians. One of Hilbert’s famously thought experiments was called about the infinity. He presented this at a lecture in 1929.

Consider a hypothetical hotel with an infinite number of rooms which are all already taken. When a finite number of new guests arrive, you can move the person in room 1 to room 2, then simultaneously move the person in room 2 to room 3, the person in room 3 to room 4 etc. moving every guest in room to room

Now consider an infinite number of new guests. By moving every guest in room to a room , we would then be left with an infinite number of odd-numbered rooms which will be free for all the infinitely new guests.

Finally, consider infinitely many coaches, with infinitely many guests each waiting to get into the hotel. To deal with the first coach, take the guest in the first room and move them to room 2, the guest in room 2 and move them to room 4, the guest in room 3 and move them to room 18. For the second coach, you would take the guest in room 1 and move them to room 3, the guest in room 2 and move them to room 9 and guest in room 3 to room 27 etc. The succession is in prime numbers. Each seat would take the seat prime number to the power of the th room. What Hilbert showed was the counter-intuitively of infinite sets. No matter how much you add to them they will be infinite.

1990 in Paris, Hilbert presented 23 problems to the international congress of mathematicians, the brightest minds in the world. He thought, if answered correctly, these problems would carry mathematics to a new level. He was correct, these problems guided mathematicians for the rest of the 20th century, three of his problems are still unsolved to this day. He later received the Bolyai prize, an international prize for mathematicians awarded every 5 years. This only confirmed what everyone already knew, which was that Hilbert was truly a great mathematician. In addition to his about infinity experiment and his famous 23 problems, Hilbert was also known for his formulated theory of Hilbert Spaces. Which is will explain in more detail later in this paper.

At the end of 1930, aged 68, Hilbert retired from his teaching position at Göttingen University. Hilbert passed away from an unknown cause at the age of 81 on the 14th of February 1943 in Göttingen. This was during the Nazi regime in Germany which caused many of his friends and colleagues to retire forced them to move away as they were Jewish. As many of them were no longer in Germany, Hilbert’s funeral was attended by under a dozen people, and his death wasn’t world known until months after On his tombstone are his famous lines he spoke at the conclusion of his retirement address to the society of German Scientists and Physicians on the 8th of September 1930; “we must know, we will know”. David Hilbert is still a legacy to this day. His contributions to mathematics have resulted in directing the subject for an entire century and created the foundations for many areas in modern mathematics.

I will now go into more detail on what a Hilbert Space is. Hilbert Spaces are a generalisation of Euclidian space. To understand some of the operations required for a Hilbert Space to exist, I will start by giving a short recap on vector spaces.