Question

Let \(X\) be a Banach space whose norm \(\|\cdot\|\) satisfies the parallelogram equality. Define \((x, y)\) by the polarization identity, and prove that \((x, y)\) is an inner product. Hint: Clearly, \(\left(\cdot_{1} \cdot\right)\) is continuous in both coordinates, \((x, y)=\overline{(y, x)}\) and \((-x, y)=-(x, y)\). Using the parallelogram equality, show that \((x+y, z)=\) \((x, z)+(y, z)\). Then by induction, \((n x, y)=n(x, y)\) for all \(n \in \mathbf{N}\), and hence also for all integers \(n\). Given \(\frac{n}{m}\), we have \(\left(\frac{n}{m} x, y\right)=n\left(\frac{1}{m} x, y\right)=\frac{n}{m} m\left(\frac{1}{m} x, y\right)=\frac{n}{m}\left(\frac{m}{m} x, y\right)=\) \(\frac{n}{m}(x, y)\). By continuity, we get \((\alpha x, y)=\alpha(x, y)\) for all \(\alpha \in \mathbf{R}\).

Short answer

\((x,y)=\frac{1}{4}(||x+y||^2-||x-y||^2)\) is an inner product due to continuity, conjugate symmetry, linearity, and homogeneity.

Step-by-step solution

Define the Inner Product using the Polarization Identity

The given hint suggests using the polarization identity to define \((x, y)\). The polarization identity in a Banach space is defined as: \((x, y) = \frac{1}{4}(||x + y||^2 - ||x - y||^2)\). For our Banach space X, define \( (x,y) \) using this polarization identity.

Show Continuity in Both Coordinates

Using the norm properties, it's straightforward to show that \((x, y) = \frac{1}{4}(||x + y||^2 - ||x - y||^2)\) is continuous in both coordinates. This is because norm is continuous and the operations involved (addition and subtraction of vectors, and squaring) are continuous.

Show Conjugate Symmetry

Next, demonstrate that \((x, y) = \overline{(y, x)}\). This follows from the symmetry properties of the norm: \((x, y) = \frac{1}{4}(||x + y||^2 - ||x - y||^2) = \overline{\frac{1}{4}(||y + x||^2 - ||y - x||^2)} = \overline{(y, x)}\).

Show Linearity in the First Coordinate

Use the given hint along with parallelogram equality to show \((x+y, z) = (x, z) + (y, z)\). The parallelogram law states: \[||x+y||^2 + ||x-y||^2 = 2||x||^2 + 2||y||^2\] and helps establish the distributivity over vector addition.

Show Homogeneity

By induction, it is claimed that \( (n x, y) = n (x, y) \) for all \ (n \in \mathbf{N})\, and hence for all integers \ n \. With rational numbers \ \frac{n}{m}, \ show \( \left(\frac{n}{m} x, y \right)= \frac{n}{m} (x, y)\). Finally, by continuity, this holds for all real numbers \ (\alpha \in \mathbf{R})\.

Conclusion

Having shown all required properties: linearity in the first coordinate, conjugate symmetry, and homogeneity, we conclude that \(( \cdot , \cdot )\) as defined by the polarization identity is indeed an inner product.

Key concepts

Banach Space

A Banach space is a special type of vector space equipped with a norm that allows for the measurement of vector lengths and distances between vectors.
This space is also complete, meaning that any Cauchy sequence of vectors in this space converges to a vector within the same space.
This property is crucial for many areas of analysis and ensures the robustness of various mathematical operations.
Another key aspect of Banach spaces is their structure, which provides a consistent foundation for discussing concepts like continuity and convergence.
The norm within a Banach space must satisfy the parallelogram equality to facilitate the inner product formation discussed in this problem.

Polarization Identity

The polarization identity is a formula that allows us to define an inner product using a norm.
For a Banach space X with norm \(...orm \), the polarization identity is given by: \((x, y) = \frac{1}{4}(||x + y||^2 - ||x - y||^2)\).
This formula essentially expresses the inner product in terms of the norm of the sum and difference of two vectors.
The beauty of the polarization identity lies in its ability to reveal inner product structure from norm-based information.
Understanding this identity is critical since it connects the geometric concept of distance (via the norm) to the algebraic concept of the inner product.

Parallelogram Equality

The parallelogram equality is an equation that norms in inner product spaces must satisfy.
It is stated as: \[||x+y||^2 + ||x-y||^2 = 2||x||^2 + 2||y||^2\].
This identity essentially characterizes the behavior of lengths of vectors under addition and subtraction.
In simpler terms, it says that the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of all four sides.
Proving that a norm satisfies the parallelogram equality is a significant step in showing that the norm can be derived from an inner product.

Continuity

Continuity in this context refers to the fact that the inner product \((x, y)\) is continuous in both coordinates.
This means if you have a sequence of vectors that converge to some vector, the inner product involving these vectors will also converge to the inner product involving the limit vector.
Formally, if \x_n \to x\ and \y_n \to y\, then \((x_n, y_n) \to (x, y)\).
The continuity of inner product is often a direct consequence of the continuity properties of the norm and the operations involved (like vector addition and scalar multiplication).
This property ensures that small changes in the vectors result in small changes in their inner product, providing stability to the calculations.

Conjugate Symmetry

Conjugate symmetry is a property of the inner product where \((x, y) = \overline{(y, x)}\).
Essentially, this means the inner product is symmetric if we take the complex conjugate.
In other words, swapping the vectors in the inner product and taking the complex conjugate yields the same value.
This property ensures that the inner product behaves consistently under the reflection of arguments.
Conjugate symmetry is crucial because it maintains the relationship between vectors when considered in complex vector spaces.
This symmetry aligns with physical and geometric intuitions about angles and lengths in spaces involving complex numbers.

Linearity

Linearity, in the context of inner products, means the inner product is linear in its first argument.
Formally, for vectors \(x,y,z\) and scalar \alpha,\ the linearity property states:
\((x + y, z) = (x, z) + (y, z)\) and \((\alpha x, y) = \alpha (x, y)\).
Linearity is essential because it ensures that the inner product distributes over vector addition and scales appropriately with scalar multiplication.
This makes the inner product align with our intuitive understanding of combining measurements from different vectors.
Without linearity, the inner product would lack the necessary algebraic structure to support many analytical techniques and applications.

Homogeneity

Homogeneity is another fundamental property of inner products, stating that the inner product scales linearly with scalar multiplication.
For a scalar \alpha\ and vectors \(x,y\), the property is expressed as:
\((\alpha x, y) = \alpha (x, y)\).
Similar to linearity, homogeneity ensures that when you scale one of the vectors by a factor, the inner product scales by the same factor.
This property is vital for compatibility between the vector space structure and the inner product.
It guarantees that the geometric interpretation of the inner product (like projections and orthogonality) remains consistent when vectors are rescaled.
Ultimately, it means the magnitude of the inner product is proportional to the magnitude of the vectors involved.