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Let's start by understanding what Fr茅chet differentiability means. A function is Fr茅chet differentiable at a point if it can be closely approximated by a linear map at that point. This means that for a given function, there should exist a linear map that approximates how the function behaves very closely.

In mathematical terms, if we have a function \( f \), it is Fr茅chet differentiable at a point \( x \) in a vector space if there exists a linear map \( A \) such that: \( \frac{||f(x+h) - f(x) - A(h)||}{||h||} \to 0 \text{ as } h \to 0 \).

For the canonical norm in \( \text{鈩搣_{1} \) space, denoted by \( ||x||_1 = \sum_{i} |x_i| \), showing Fr茅chet differentiability involves checking if there is such a linear map that can approximate the norm function at every point. Due to the nature of the absolute value function \( |x_i| \), particularly around 0, we find that the \( \text{鈩搣_{1} \) norm is nowhere Fr茅chet differentiable.

This is because the absolute value function is not differentiable at 0, making it impossible to find a consistent linear approximation that works across the entire space. Thus, the canonical norm in \( \text{鈩搣_{1} \) space is nowhere Fr茅chet differentiable.

Now, let's move on to G芒teaux differentiability. A function is G芒teaux differentiable at a point if the directional derivative exists in every direction. This means that for a function to be G芒teaux differentiable at a point \( x \), the derivative in the direction of any vector \( h \) should exist.

Formally, for a function \( f \) to be G芒teaux differentiable at \( x \), it should satisfy: \(\frac{d}{dt} f(x + th)|_{t=0} \) for all vectors \( h \).

When considering the canonical norm in \( \text{鈩搣_{1} \) space, i.e., \( ||x||_1 = \sum_{i} |x_i| \), we check if the directional derivatives exist. Given that G芒teaux differentiability at a point \( x = (x_i) \) requires each \( x_i \) not equal to zero, the absolute value behaves smoothly away from zero. Thus, the canonical norm in \( \text{鈩搣_{1} \) space is G芒teaux differentiable at \( x \) if and only if \( x_i eq 0 \) for every \( i \).

In essence, if none of the components of the vector \( x \) are zero, then the norm function behaves well enough in every direction to be considered G芒teaux differentiable.

The canonical norm in \( \text{鈩搣_{1} \) space is a fundamental concept in functional analysis. It's defined for a vector \( x = (x_i) \) in \( \text{鈩搣_{1} \) space by the sum of the absolute values of its components: \( ||x||_1 = \sum_{i} |x_i| \).

This norm measures the 'length' or 'size' of the vector in a specific way by summing up the magnitudes of all its components. It's called the canonical norm because it is a standard way to define the norm in this space.

Understanding the behavior of the canonical norm is crucial for studying differentiability properties. In particular, challenges in differentiability arise around zero due to the non-differentiable nature of the absolute value function at zero.

For example, when checking for G芒teaux differentiability of the canonical norm, we see that issues occur when any component \( x_i \) of the vector is zero. The smooth behavior away from zero ensures differentiability in those regions, but not at or around zero.

Lastly, let's touch on the concept of a vector space, a central building block in functional analysis. A vector space is a collection of vectors, which can be added together and multiplied (scaled) by numbers, called scalars.

In more formal terms, a vector space over a field \( F \) (like the field of real numbers \( \text{鈩潁 \)) is a set \( V \) equipped with two operations: vector addition and scalar multiplication. These operations must satisfy certain axioms such as associativity, commutativity, and distribution.

To give a concrete example, the set of all sequences of real numbers that converge to zero, denoted by \( \text{鈩搣_{1} \), forms a vector space. Vectors in \( \text{鈩搣_{1} \) can be added together, and scaled by real numbers, while satisfying the necessary axioms.

In this context, we often study norms like the canonical norm to measure the size or length of vectors within this space, thus allowing us to analyze properties such as convergence, continuity, and differentiability.