# About bases of complex (Hilbert) space

• KFC
In summary, the conversation focused on finding a basis for 3-dimensional complex (Hilbert) space. It was suggested that the simplest basis would be the same as the canonical basis for 3-dimensional real linear space, with the entries being complex numbers. The conversation also discussed the possibility of constructing an orthonormal basis for complex space.
KFC
Hi there,
In 3-dimensional real linear space, the simplest bases can be taken as the canonical bases

$$\hat{x} = \left(\begin{matrix}1 \\ 0 \\ 0\end{matrix}\right), \qquad \hat{y} = \left(\begin{matrix}0 \\ 1 \\0\end{matrix}\right), \qquad \hat{z} = \left(\begin{matrix}0 \\ 0 \\ 1\end{matrix}\right)$$

I wonder what's the simplest counterpart for 3-dimensional in complex (hilbert) space?

the basis you gave for R^3 is also a basis for C^3

Thanks. But can we construct a basis similar to those for R^3 but with the entries be complex? and how?

The exact same basis will do. Remember that your scalars are complex numbers.

KFC said:
Thanks. But can we construct a basis similar to those for R^3 but with the entries be complex? and how?

sure, if you replace your 1's with i's it's still a basis. all you have to do is pick 3 vectors so that no one of them is a linear combination of the other 2. if you want it to be an orthonormal basis (which is usually useful) then you need to also make sure that each vector is of unit length and orthogonal to the other 2.

## 1. What is a complex (Hilbert) space?

A complex (Hilbert) space is a mathematical concept that extends the idea of a Euclidean space to include complex numbers. It is a vector space equipped with an inner product that allows for the measurement of angles and distances between vectors. This space is essential in many areas of mathematics, including functional analysis, quantum mechanics, and signal processing.

## 2. What are the basic properties of a complex (Hilbert) space?

Some of the key properties of a complex (Hilbert) space include linearity, completeness, and orthogonality. Linearity means that the space is closed under vector addition and scalar multiplication. Completeness means that every Cauchy sequence in the space converges to a point within the space. Orthogonality refers to the perpendicularity of two vectors in the space with respect to the inner product.

## 3. How is a complex (Hilbert) space different from a real Hilbert space?

A complex (Hilbert) space differs from a real Hilbert space in that it allows for the use of complex numbers instead of just real numbers. This allows for a more flexible and powerful mathematical framework, as complex numbers can represent both magnitude and direction. In contrast, real Hilbert spaces only deal with magnitudes.

## 4. What are some applications of complex (Hilbert) spaces?

Complex (Hilbert) spaces have a wide range of applications in mathematics, physics, and engineering. They are used in the study of quantum mechanics, where they provide a mathematical foundation for the description of physical systems. They are also used in the analysis of signals and systems in signal processing and in the study of partial differential equations in mathematical physics.

## 5. How are bases of complex (Hilbert) spaces used in practical applications?

Bases of complex (Hilbert) spaces are essential in practical applications because they provide a convenient way to represent vectors in the space. They allow for the decomposition of a vector into a linear combination of basis vectors, which simplifies calculations and makes it easier to work with complex vectors. Bases are also used in the construction of orthogonal projections, which are used in data compression and image processing.

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